How many questions from Limits and Continuity appear in JEE?

Limits and Continuity has appeared in 149 of the last 186 JEE Main and JEE Advanced papers — about 80% of them — contributing 206 questions in total across those papers.

Is Limits and Continuity an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 80% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Limits and Continuity questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Limits and Continuity JEE Previous Year Questions with Answers

Q1·MathematicsNumericalJEE Advanced 2026

For a real number

\alpha

, let

[\alpha]

denote the greatest integer less than or equal to

\alpha

. For a finite set

S

, let

|S|

denote the number of elements in the set

S

. Consider the functions

f : (-3, 3) \to (-\infty, \infty)

and

g : (-3, 3) \to (-\infty, \infty)

defined by

f(x) = [x^{3}] \log_{e}(1 + \sin^{2}(\pi(x - [x])))

and

g(x) = x^{3} \sin^{2}(\pi \log_{e}(1 + x - [x]))

. Let

A = \{x \in (-3, 3) : f

is discontinuous at

x\}

and

B = \{x \in (-3, 3) : g

is discontinuous at

x\}

. Then the value of

|A| + 2|B| - |A \cap B|

is _____.

Q2·MathematicsSingle correctJEE Main 2026

Let

f(x) = \begin{cases} \dfrac{1}{3}, & x \le \pi/2 \\ \dfrac{b(1 - \sin x)}{(\pi - 2x)^2}, & x > \pi/2 \end{cases}

. If

f

is continuous at

x = \pi/2

, then the value of

\displaystyle\int_{0}^{3b-6} |x^2 + 2x - 3|\,dx

is:

(A) $5$
(B) $2$
(C) $3$
(D) $4$

Q3·MathematicsNumericalJEE Main 2026

Let

f(x) = \begin{cases} x^{3} + 8; & x < 0 \\ x^{2} - 4; & x \ge 0 \end{cases}

and

g(x) = \begin{cases} (x - 8)^{1/3}; & x < 0 \\ (x + 4)^{1/2}; & x \ge 0 \end{cases}

. Then the number of points, where the function

g \circ f

is discontinuous, is __________.

Q4·MathematicsSingle correctJEE Main 2026

The value of

\lim_{x \to 0}\left(\frac{x^2 \sin^2 x}{x^2 - \sin^2 x}\right)

is:

(A)2
(B)3
(C)4
(D)6

Q5·MathematicsSingle correctJEE Main 2026

Let

f(x) = \lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^{3}}

. Then the number of solutions of the equation f(x) = sin x, x ∈ ℝ is :

(A)0
(B)2
(C)3
(D)1

Q6·MathematicsSingle correctJEE Main 2026

The product of all possible values of

\alpha

, for which

\lim\limits_{x \to 0} \left( \frac{1 - \cos(\alpha x)\cos((\alpha + 1)x)\cos((\alpha + 2)x)}{\sin^2((\alpha + 1)x)} \right) = 2

, is:

(A) $-2$
(B)1
(C) $-1$
(D) $\frac{5}{4}$

Q7·MathematicsSingle correctJEE Main 2026

If

\lim_{x \to 2} \frac{\sin\left(x^{3} - 5x^{2} + ax + b\right)}{\left(\sqrt{x-1} - 1\right)\log_{e}(x-1)} = m

, then

a + b + m

is equal to :

(A)5
(B)6
(C)8
(D)10

Q8·MathematicsNumericalJEE Main 2026

The number of points in the interval

[2, 4]

, at which the function

f(x) = \left[x^2 - x - \frac{1}{2}\right]

, where

[\cdot]

denotes the greatest integer function, is discontinuous, is ________.

Q9·MathematicsSingle correctJEE Main 2026

Let

f(x) = \lim\limits_{\theta \to 0}\left(\dfrac{\cos \pi x - x^{\left(\frac{2}{\theta}\right)}\sin(x-1)}{1 + x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R

. Consider the following two statements : (I) f(x) is discontinous at x = 1. (II) f(x) is continous at x = - 1. Then,

(A)Neither (I) nor (II) is True
(B)Both (I) and (II) are True
(C)Only (II) is True
(D)Only (I) is True

Q10·MathematicsSingle correctJEE Main 2026

The value of

\lim_{x \to 0} \dfrac{\log_e\left(\sec(ex).\sec\left(e^{2}x\right)\cdot ... \cdot \sec\left(e^{10}x\right)\right)}{e^{2} - e^{2\cos x}}

is equal to

(A) $\dfrac{(e^{10}-1)}{2e^{2}(e^{2}-1)}$
(B) $\dfrac{(e^{20}-1)}{2e^{2}(e^{2}-1)}$
(C) $\dfrac{(e^{20}-1)}{2(e^{2}-1)}$
(D) $\dfrac{(e^{10}-1)}{2(e^{2}-1)}$

Q11·MathematicsSingle correctJEE Main 2026

Let [t] denote the greatest integer less than or equal to t. If the function

f(x) = \begin{cases} b^{2} \sin\left( \dfrac{\pi}{2} \left[ \dfrac{\pi}{2} (\cos x + \sin x) \cos x \right] \right) & , x < 0 \\ \dfrac{\sin x - \dfrac{1}{2}\sin 2x}{x^{3}} & , x > 0 \\ a & , x = 0 \end{cases}

is continuous at

x = 0

, then

a^{2} + b^{2}

is equal to

(A) $\dfrac{5}{8}$
(B) $\dfrac{9}{16}$
(C) $\dfrac{3}{4}$
(D) $\dfrac{1}{2}$

Q12·MathematicsSingle correctJEE Main 2026

If the function f(x)

= \frac{e^{x}\left(e^{\tan x-x}-1\right)+\log_{e}(\sec x+\tan x)-x}{\tan x-x}

is Continuous at x = 0, then the value of f(0) is equal to

(A) $2$
(B) $\frac{2}{3}$
(C) $\frac{1}{2}$
(D) $\frac{3}{2}$

Q13·MathematicsSingle correctJEE Main 2026

If

f(x)=\begin{cases} \dfrac{a|x|+x^2-2(\sin|x|)(\cos|x|)}{x} & ,\ x \neq 0 \\[6pt] b & ,\ x = 0 \end{cases}

is continuous at

x = 0

, then

a + b

is equal to :

(A)1
(B)2
(C)0
(D)4

Q14·MathematicsSingle correctJEE Main 2026

Let

f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \ne -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases}

be continuous at

x = -\frac{3}{2}

. If

fof(x) = \frac{7}{5}

, then x is equal to :

(A)2
(B)1
(C)0
(D)1.4

Q15·MathematicsSingle correctJEE Main 2026

If

\lim\limits_{x \to 0} \frac{e^{(a-1)x} + 2\cos bx + (c-2)e^{-x}}{x \cos x - \log_{e}(1+x)} = 2

, then

a^{2} + b^{2} + c^{2}

is equal to :

(A)5
(B)3
(C)7
(D)9

Q16·MathematicsSingle correctJEE Main 2026

Let

f : R \to (0, \infty)

be a twice differentiable function such that

f(3) = 18

,

f'(3) = 0

and

f''(3) = 4

. Then

\lim_{x \to 1}\left( \log_{e}\left( \frac{f(2+x)}{f(3)} \right)^{\frac{18}{(x-1)^{2}}} \right)

is equal to :

(A)1
(B)9
(C)2
(D)18

Q17·MathematicsNumericalJEE Main 2026

Let

[\cdot]

denote the greatest integer function and

f(x) = \lim_{n \to \infty} \frac{1}{n^{3}} \sum_{k=1}^{n} \left[\frac{k^{2}}{3^{x}}\right]

. Then

12\sum_{j=1}^{\infty} f(j)

is equal to _______.

Q18·MathematicsSingle correctJEE Advanced 2025

Let

x_0

be the real number such that

e^{x_0} + x_0 = 0

. For a given real number α, define

g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3\left(e^x + 1\right)}

for all real numbers x. Then which one of the following statements is TRUE ?

(A)For α = 2, $\lim_{x \rightarrow x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 0$
(B)For α = 2, $\lim_{x \rightarrow x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 1$
(C)For α = 3, $\lim_{x \rightarrow x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 0$
(D)For α = 3, $\lim_{x \rightarrow x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = \frac{2}{3}$

Q19·MathematicsNumericalJEE Advanced 2025

Let α and β be the real numbers such that

\lim_{x \to 0} \frac{1}{x^3}\left( \frac{\alpha}{2} \int_0^x \frac{1}{1-t^2} dt + \beta x \cos x \right) = 2

. Then the value of α + β is _______ .

Q20·MathematicsSingle correctJEE Advanced 2025

Let R denote the set of all real numbers. For a real number x, let [x] denote the greatest integer less than or equal to x. Let n denote a natural number. Match each entry in List-I to the correct entries in List-II and choose the correct option. The correct option is:

List-I	List-II
P.The minimum value of n for which the function $f(x) = \left[\frac{10x^3 - 45x^2 + 60x + 35}{n}\right]$ is continuous on the interval [1, 2], is	1.8
Q.The minimum value of n for which $g(x) = (2n^2 - 13n - 15)(x^3 + 3x)$ , $x \in R$ , is an increasing function on R, is	2.9
R.The smallest natural number n which is greater than 5, such that $x = 3$ is a point of local minima of $h(x) = (x^2 - 9)^n(x^2 + 2x + 3)$ , is	3.5
S.Number of $x_0 \in R$ such that $l(x) = \sum_{k=0}^{4}\left( \sin\|x - k\| + \cos\left\|x - k + \frac{1}{2}\right\| \right)$ , $x \in R$ , In NOT differentiable at $x_0$ is	4.6
	5.10

(A)(P) → (1), (Q) → (3), (R) → (2), (S) → (5)
(B)(P) → (2), (Q) → (1), (R) → (4), (S) → (3)
(C)(P) → (5), (Q) → (1), (R) → (4), (S) → (3)
(D)(P) → (2), (Q) → (3), (R) → (1), (S) → (5)

Q21·MathematicsSingle correctJEE Main 2025

Given below are two statements: Statement I:

\lim\limits_{x\to0}\dfrac{\tan^{-1}x+\log_e\sqrt{\dfrac{1+x}{1-x}}-2x}{x^5}=\dfrac{2}{5}

. Statement II:

\lim\limits_{x\to1}x^{\frac{2}{1-x}}=\dfrac{1}{e^2}

. In the light of the above statements, choose the correct answer from the options given below:

(A)Statement I is false but Statement II is true
(B)Statement I is true but Statement II is false
(C)Both Statement I and Statement II are false
(D)Both Statement I and Statement II are true

Q22·MathematicsIntegerJEE Main 2025

The number of points of discontinuity of the function

f(x)=\left[\dfrac{x^2}{2}\right]-\left[\sqrt{x}\right]

,

x\in[0,4]

, where

[\cdot]

denotes the greatest integer function is ______.

Q23·MathematicsSingle correctJEE Main 2025

\displaystyle\lim_{x\to0^+}\dfrac{\tan\left(5(x)^{1/3}\right)\log_e(1+3x^2)}{\left(\tan^{-1}3\sqrt{x}\right)^2\left(e^{5(x)^{4/3}}-1\right)}

is equal to

(A) $\dfrac{1}{15}$
(B)1
(C) $\dfrac{1}{3}$
(D) $\dfrac{5}{3}$

Q24·MathematicsIntegerJEE Main 2025

If the function

f(x)=\dfrac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}

is continuous at

x=0

, then

f(0)

is equal to __________.

Q25·MathematicsSingle correctJEE Main 2025

Let

f:\mathbb{R}\to\mathbb{R}

be a polynomial function of degree four having extreme values at

x=4

and

x=5

. If

\lim\limits_{x\to 0}\dfrac{f(x)}{x^2}=5

, then

f(2)

is equal to:

(A)12
(B)10
(C)8
(D)14

Q26·MathematicsIntegerJEE Main 2025

For

t>-1

, let

\alpha_t

and

\beta_t

be the roots of the equation

\left((t+2)^{1/7}-1\right)x^2+\left((t+2)^{1/6}-1\right)x+\left((t+2)^{1/21}-1\right)=0

. If

\lim\limits_{t\to -1^+}\alpha_t=a

and

\lim\limits_{t\to -1^+}\beta_t=b

, then

72(a+b)^2

is equal to __________.

Q27·MathematicsSingle correctJEE Main 2025

If

\displaystyle\lim_{x\to1}\dfrac{(x-1)(6+\lambda\cos(x-1))+\mu\sin(1-x)}{(x-1)^3}=-1

, where

\lambda,\mu\in\mathbb{R}

, then

\lambda+\mu

is equal to

(A)18
(B)20
(C)19
(D)17

Q28·MathematicsSingle correctJEE Main 2025

Let

f

be a differentiable function on

\mathbb{R}

such that

f(2)=1

,

f'(2)=4

. Let

\lim_{x\to0}\left(\dfrac{f(2+x)}{f(2)}\right)^{3/x}=e^{\alpha}

. Then the number of times the curve

y=4x^3-4x^2-4(\alpha-7)x-\alpha

meets x-axis is:

(A)2
(B)1
(C)0
(D)3

Q29·MathematicsIntegerJEE Main 2025

Let

m

and

n

be the number of points at which the function

f(x)=\max\{x,x^3,x^5,\ldots,x^{21}\}

,

x\in\mathbb{R}

, is not differentiable and not continuous, respectively. Then

m+n

is equal to ______.

Q30·MathematicsSingle correctJEE Main 2025

Let

f(x)=\begin{cases}(1+ax)^{1/x}, & x<0\\ 1+b, & x=0\\ \dfrac{(x+4)^{1/2}-2}{(x+c)^{1/3}-2}, & x>0\end{cases}

be continuous at

x=0

. Then

e^a\cdot b\cdot c

is equal to:

(A)64
(B)72
(C)48
(D)36

Q31·MathematicsIntegerJEE Main 2025

If

\displaystyle\lim_{x\to 0}\left(\dfrac{\tan x}{x}\right)^{1/x^2}=p

, then

96\log_e p

is equal to __________.

Q32·MathematicsSingle correctJEE Main 2025

For

\alpha,\beta,\gamma\in R

, if

\displaystyle\lim_{x\to0}\dfrac{x^2\sin\alpha x+(\gamma-1)e^{x^2}}{\sin 2x-\beta x}=3

, then

\beta+\gamma-\alpha

is equal to:

(A)7
(B)4
(C)6
(D) $-1$

Q33·MathematicsSingle correctJEE Main 2025

If

\displaystyle\lim_{x\to0}\dfrac{\cos(2x)+a\cos(4x)-b}{x^4}

is finite, then

(a+b)

is equal to:

(A) $\dfrac{1}{2}$
(B)0
(C) $\dfrac{3}{4}$
(D) $-1$

Q34·MathematicsIntegerJEE Main 2025

Let

[t]

be the greatest integer less than or equal to t. Then the least value of

p\in N

for which

\displaystyle\lim_{x\to 0^+}\left(x\left(\left[\dfrac{1}{x}\right]+\left[\dfrac{2}{x}\right]+\cdots+\left[\dfrac{p}{x}\right]\right)-x^2\left(\left[\dfrac{1^2}{x^2}\right]+\left[\dfrac{2^2}{x^2}\right]+\cdots+\left[\dfrac{9^2}{x^2}\right]\right)\right)\ge 1

is equal to ______.

Q35·MathematicsIntegerJEE Main 2025

If

\displaystyle\lim_{t\to 0}\left(\int_0^1 (3x+5)^t\,dx\right)^{\frac{1}{t}}=\dfrac{\alpha}{5e}\left(\dfrac{8}{5}\right)^{\frac{2}{3}}

, then

\alpha

is equal to ______.

Q36·MathematicsIntegerJEE Main 2025

Let

f(x)=\lim_{n\to\infty}\sum_{r=0}^{n}\left(\frac{\tan(x/2^{r+1})+\tan^3(x/2^{r+1})}{1-\tan^2(x/2^{r+1})}\right)

. Then

\lim_{x\to0}\frac{e^x-e^{f(x)}}{x-f(x)}

is equal to ______.

Q37·MathematicsIntegerJEE Main 2025

Let

f(x)=\begin{cases}3x, & x<0\\ \min\{1+x+[x], x+2[x]\}, & 0\le x\le 2\\ 5, & x>2\end{cases}

where

[.]

denotes greatest integer function. If

\alpha

and

\beta

are the number of points, where f is not continuous and is not differentiable, respectively, then

\alpha+\beta

equals____.

Q38·MathematicsSingle correctJEE Main 2025

Consider the region

R=\left\{(x,y):x\le y\le 9-\dfrac{11}{3}x^2,\ x\ge 0\right\}

. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R, is :

(A) $\dfrac{625}{111}$
(B) $\dfrac{730}{119}$
(C) $\dfrac{567}{121}$
(D) $\dfrac{821}{123}$

Q39·MathematicsSingle correctJEE Main 2025

Let

f:\mathbb{R}-\{0\}\to\mathbb{R}

be a function such that

f(x)-6f\left(\dfrac{1}{x}\right)=\dfrac{35}{3x}-\dfrac{5}{2}

. If

\lim_{x\to 0}\left(\dfrac{1}{\alpha x}+f(x)\right)=\beta

;

\alpha,\beta\in\mathbb{R}

, then

\alpha+2\beta

is equal to

(A) $3$
(B) $5$
(C) $4$
(D) $6$

Q40·MathematicsSingle correctJEE Main 2025

\lim_{x\to 0}\operatorname{cosec}x\left(\sqrt{2\cos^2 x+3\cos x}-\sqrt{\cos^2 x+\sin x+4}\right)

is

(A) $0$
(B) $\dfrac{1}{2\sqrt5}$
(C) $\dfrac{1}{\sqrt{15}}$
(D) $-\dfrac{1}{2\sqrt5}$

Q41·MathematicsSingle correctJEE Main 2025

Let

[x]

denote the greatest integer function, and let

m

and

n

respectively be the numbers of the points, where the function

f(x)=[x]+|x-2|

,

-2<x<3

, is not continuous and not differentiable. Then

m+n

is equal to:

(A)6
(B)9
(C)8
(D)7

Q42·MathematicsSingle correctJEE Main 2025

If the function

f(x)=\begin{cases}\dfrac{2}{x}\{\sin(k_1+1)x+\sin(k_2-1)x\}, & x<0\\[2mm] 4, & x=0\\[2mm] \dfrac{2}{x}\log_e\left(\dfrac{2+k_1x}{2+k_2x}\right), & x>0\end{cases}

is continuous at

x=0

, then

k_1^2+k_2^2

is equal to

(A) $8$
(B) $20$
(C) $5$
(D) $10$

Q43·MathematicsIntegerJEE Main 2025

If the set of all values of

a

, for which the equation

5x^3-15x-a=0

has three distinct real roots, is the interval

(\alpha,\beta)

, then

\beta-2\alpha

is equal to _______

Q44·MathematicsSingle correctJEE Main 2025

\displaystyle\lim_{x\to\infty}\dfrac{\left(2x^2-3x+5\right)\left(3x-1\right)^{x/2}}{\left(3x^2+5x+4\right)\sqrt{\left(3x+2\right)^x}}

is equal to :

(A) $\dfrac{2}{\sqrt{3e}}$
(B) $\dfrac{2e}{\sqrt{3}}$
(C) $\dfrac{2e}{3}$
(D) $\dfrac{2}{3\sqrt{e}}$

Q45·MathematicsSingle correctJEE Main 2025

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is

1

cm, the ice-cream melts at the rate of

81\,\text{cm}^3/\text{min}

and the thickness of the ice-cream layer decreases at the rate of

\dfrac{1}{4\pi}

cm/min. The surface area (in

\text{cm}^2

) of the chocolate ball (without the ice-cream layer) is :

(A) $225\pi$
(B) $128\pi$
(C) $196\pi$
(D) $256\pi$

Q46·MathematicsSingle correctJEE Main 2025

If

\displaystyle\lim_{x\to\infty}\left(\dfrac{e}{1-e}\left(\dfrac{1}{e}-\dfrac{x}{1+x}\right)\right)^x=\alpha

, then the value of

\dfrac{\log_e\alpha}{1+\log_e\alpha}

equals:

(A)e
(B) $e^{-2}$
(C) $e^{2}$
(D) $e^{-1}$

Q47·MathematicsSingle correctJEE Advanced 2024

Let

k \in \mathbb{R}

. If

\lim_{x \to 0^+} \left(\sin(\sin kx) + \cos x + x\right)^{\frac{2}{x}} = e^6

, then the value of

k

is

(A)1
(B)2
(C)3
(D)4

Q48·MathematicsMultiple correctJEE Advanced 2024

Let

S

be the set of all

(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}

such that

\lim_{x \to \infty} \frac{\sin\left(x^2\right)\left(\log_e x\right)^{\alpha} \sin\left(\frac{1}{x^2}\right)}{x^{\alpha\beta}\left(\log_e (1+x)\right)^{\beta}} = 0

. Then which of the following is(are) correct ?

(A) $(-1, 3) \in S$
(B) $(-1, 1) \in S$
(C) $(1, -1) \in S$
(D) $(1, -2) \in S$

Q49·MathematicsNumericalJEE Main 2024

Let

f:(0, \pi) \to \mathbb{R}

be a function given by

f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ (1 + |\cot x|)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi \end{cases}

Where

a, b \in \mathbb{Z}

. If

f

is continuous at

x = \frac{\pi}{2}

, then

a^2 + b^2

is equal to ________.

Q50·MathematicsSingle correctJEE Main 2024

\displaystyle\lim_{x\to\frac{\pi}{2}}\dfrac{\int_{x^{3}}^{(\pi/2)^{3}}\left(\sin\left(2t^{1/3}\right)+\cos\left(t^{1/3}\right)\right)dt}{\left(x-\dfrac{\pi}{2}\right)^{2}}

is equal to:

(A) $\dfrac{9\pi^{2}}{8}$
(B) $\dfrac{11\pi^{2}}{10}$
(C) $\dfrac{3\pi^{2}}{2}$
(D) $\dfrac{4\pi^{2}}{9}$

Q51·MathematicsSingle correctJEE Main 2024

\displaystyle\lim_{x\to 0}\dfrac{e-(1+2x)^{\frac{1}{2x}}}{x}

is equal to:

(A) $e$
(B) $\dfrac{-2}{e}$
(C) $0$
(D) $-e^{2}$

Q52·MathematicsSingle correctJEE Main 2024

For

a,b>0

, let

f(x)=\begin{cases}\dfrac{\tan((a+1)x)+b\tan x}{x} & ,\ x<0\\[2pt] 3 & ,\ x=0\\[2pt] \dfrac{\sqrt{ax+b^2x^2}-\sqrt{ax}}{b\sqrt{a}\,x\sqrt{x}} & ,\ x>0\end{cases}

be a continuous function at

x=0

. Then

\dfrac{b}{a}

is equal to

(A) $5$
(B) $4$
(C) $8$
(D) $6$

Q53·MathematicsNumericalJEE Main 2024

If

\alpha=\displaystyle\lim_{x\to 0^+}\left(\dfrac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)

and

\beta=\displaystyle\lim_{x\to 0}(1+\sin x)^{\frac{1}{2}\cot x}

are the roots of the quadratic equation

ax^2+bx-\sqrt{e}=0

, then

12\log_e(a+b)

is equal to _______ .

Q54·MathematicsNumericalJEE Main 2024

The value of

\displaystyle\lim_{x\to0}2\left(\dfrac{1-\sqrt{\cos 2x}\,\sqrt[3]{\cos 3x}\cdots\sqrt[10]{\cos 10x}}{x^2}\right)

is ___

Q55·MathematicsSingle correctJEE Main 2024

\lim_{n\to\infty} \frac{(1^2-1)(n-1) + (2^2-2)(n-2) + \cdots + ((n-1)^2-(n-1))\cdot 1}{(1^3 + 2^3 + \cdots + n^3) - (1^2 + 2^2 + \cdots + n^2)}

is equal to:

(A) $\tfrac{2}{3}$
(B) $\tfrac{1}{3}$
(C) $\tfrac{3}{4}$
(D) $\tfrac{1}{2}$

Q56·MathematicsSingle correctJEE Main 2024

If the function

f(x) = \dfrac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}

,

x \in \mathbb{R}

, is continuous at

x = 0

, then

f(0)

is equal to:

(A) $2$
(B) $-2$
(C) $4$
(D) $-4$

Q57·MathematicsSingle correctJEE Main 2024

Let

f:[-1,2]\to\mathbb{R}

be given by

f(x)=2x^2+x+[x^2]-[x]

, where

[t]

denotes the greatest integer less than or equal to

t

. The number of points where

f

is not continuous is:

(A)6
(B)3
(C)4
(D)5

Q58·MathematicsNumericalJEE Main 2024

Let

a>0

be a root of the equation

2x^2+x-2=0

. If

\displaystyle\lim_{x\to\frac{1}{a}}\dfrac{16\left(1-\cos(2+x-2x^2)\right)}{(1-ax)^2}=\alpha+\beta\sqrt{17}

, where

\alpha,\beta\in\mathbb{Z}

, then

\alpha+\beta

is equal to __________.

Q59·MathematicsNumericalJEE Main 2024

If

\displaystyle\lim_{x\to1}\dfrac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\dfrac{m\sqrt5}{n(2n)^{2/3}}

, where

\gcd(m,n)=1

, then

8m+12n

is equal to ___

Q60·MathematicsSingle correctJEE Main 2024

Let

f(x)=\dfrac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}

for

x\ne 0

, and

f(0)=a\,\log_e 2\,\log_e 3

. If

f

is continuous at

x=0

, then the value of

a^2

is equal to

(A)768
(B)1152
(C)746
(D)1250

Q61·MathematicsSingle correctJEE Main 2024

Let

f(x)=\displaystyle\int_0^x\left(t+\sin\left(1-e^t\right)\right)dt

,

x\in\mathbb{R}

. Then

\displaystyle\lim_{x\to 0}\dfrac{f(x)}{x^3}

is equal to

(A) $\tfrac{1}{6}$
(B) $-\tfrac{1}{6}$
(C) $-\tfrac{2}{3}$
(D) $\tfrac{2}{3}$

Q62·MathematicsSingle correctJEE Main 2024

Let

f:\mathbb{R}\to\mathbb{R}

be a function given by

f(x)=\begin{cases}\dfrac{1-\cos 2x}{x^2} & ,\ x<0\\ \alpha & ,\ x=0\\ \dfrac{\beta\sqrt{1-\cos x}}{x} & ,\ x>0\end{cases}

where

\alpha,\beta\in\mathbb{R}

. If

f

is continuous at

x=0

, then

\alpha^2+\beta^2

is equal to:

(A) $48$
(B) $12$
(C) $3$
(D) $6$

Q63·MathematicsNumericalJEE Main 2024

Let

\{x\}

denote the fractional part of

x

and

f(x)=\dfrac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}

,

x\ne0

. If

L

and

R

respectively denote the left hand limit and the right hand limit of

f(x)

at

x=0

, then

\dfrac{32}{\pi^2}(L^2+R^2)

is equal to ___

Q64·MathematicsSingle correctJEE Main 2024

Let

f(x)=\big|\,2x^2+5|x|-3\,\big|

,

x\in\mathbb R

. If

m

and

n

denote the number of points where

f

is not continuous and not differentiable respectively, then

m+n

is equal to:

(A)5
(B)2
(C)0
(D)3

Q65·MathematicsNumericalJEE Main 2024

If

y=\dfrac{(\sqrt x+1)(x^2-\sqrt x)}{x\sqrt x+x+\sqrt x}+\dfrac{1}{15}(3\cos^2 x-5)\cos^3 x

, then

96\,y'\!\left(\dfrac{\pi}{6}\right)

is equal to __________.

Q66·MathematicsSingle correctJEE Main 2024

Let

f:\mathbb{R}\to\mathbb{R}

be defined as

f(x)=\begin{cases}\dfrac{a-b\cos 2x}{x^2} & ,\ x<0\\ x^2+cx+2 & ,\ 0\le x\le1\\ 2x+1 & ,\ x>1\end{cases}

. If

f

is continuous everywhere in

\mathbb{R}

and

m

is the number of points where

f

is NOT differentiable then

m+a+b+c

equals:

(A) $1$
(B) $4$
(C) $3$
(D) $2$

Q67·MathematicsSingle correctJEE Main 2024

Let

f(x)=\begin{cases} x-1,& x\text{ is even},\\ 2x,& x\text{ is odd},\end{cases}\;x\in\mathbb N

. If for some

a\in\mathbb N

,

f(f(f(a)))=21

, then

\displaystyle\lim_{x\to a^-}\left\{\dfrac{|x|^3}{a}-\left[\dfrac{x}{a}\right]\right\}

, where

[t]

denotes the greatest integer less than or equal to

t

, is equal to:

(A)121
(B)144
(C)169
(D)225

Q68·MathematicsSingle correctJEE Main 2024

Let

g(x)

be a linear function and

f(x)=\begin{cases}g(x), & x\le 0\\ \left(\dfrac{1+x}{2+x}\right)^{\frac{1}{x}}, & x>0\end{cases}

, is continuous at

x=0

. If

f'(1)=f(-1)

, then the value of

g(3)

is

(A) $\dfrac{1}{3}\log_e\left(\dfrac{4}{9e^{1/3}}\right)$
(B) $\dfrac{1}{3}\log_e\left(\dfrac{4}{9}\right)+1$
(C) $\log_e\left(\dfrac{4}{9}\right)-1$
(D) $\log_e\left(\dfrac{4}{9e^{1/3}}\right)$

Q69·MathematicsSingle correctJEE Main 2024

Consider the function

f:(0,\infty)\to R

defined by

f(x)=e^{-\left|\log_e x\right|}

. If

m

and

n

be respectively the number of points at which

f

is not continuous and

f

is not differentiable, then

m+n

is

(A)0
(B)3
(C)1
(D)2

Q70·MathematicsSingle correctJEE Main 2024

\displaystyle\lim_{x\to 0}\dfrac{e^{2|\sin x|}-2|\sin x|-1}{x^2}

(A)is equal to $-1$
(B)does not exist
(C)is equal to $1$
(D)is equal to $2$

Q71·MathematicsNumericalJEE Main 2024

If

\displaystyle\lim_{x\to0}\dfrac{ax^2e^x-b\log_e(1+x)+cxe^{-x}}{x^2\sin x}=1

, then

16(a^2+b^2+c^2)

is equal to ______.

Q72·MathematicsSingle correctJEE Main 2024

If the function

f:(-\infty,-1]\to(a,b]

defined by

f(x)=e^{x^3-3x+1}

is one-one and onto, then the distance of the point

P(2b+4,\ a+2)

from the line

x+e^{-3}y=4

is

(A) $2\sqrt{1+e^6}$
(B) $4\sqrt{1+e^6}$
(C) $3\sqrt{1+e^6}$
(D) $\sqrt{1+e^6}$

Q73·MathematicsSingle correctJEE Main 2024

Let

f:R\to(0,\infty)

be strictly increasing function such that

\displaystyle\lim_{x\to\infty}\dfrac{f(7x)}{f(x)}=1

. Then the value of

\displaystyle\lim_{x\to\infty}\left[\dfrac{f(5x)}{f(x)}-1\right]

is equal to

(A)4
(B)0
(C) $\dfrac{7}{5}$
(D)1

Q74·MathematicsSingle correctJEE Main 2024

Let

f:R-\{0\}\to R

be a function satisfying

f\left(\dfrac{x}{y}\right)=\dfrac{f(x)}{f(y)}

for all

x,y

,

f(y)\ne 0

. If

f'(1)=2024

, then:

(A) $xf'(x)-2024f(x)=0$
(B) $xf'(x)+2024f(x)=0$
(C) $xf'(x)+f(x)=2024$
(D) $xf'(x)-2023f(x)=0$

Q75·MathematicsSingle correctJEE Main 2024

Let

f:\left[-\dfrac\pi2,\dfrac\pi2\right]\to\mathbb{R}

be a differentiable function such that

f(0)=\dfrac12

. If the

\displaystyle\lim_{x\to0}\dfrac{x\displaystyle\int_0^x f(t)\,dt}{e^{x^2}-1}=\alpha

, then

8\alpha^2

is equal to:

(A) $16$
(B) $2$
(C) $1$
(D) $4$

Q76·MathematicsSingle correctJEE Main 2024

The function

f(x)=\dfrac{x}{x^2-6x-16},\ x\in\mathbb{R}-\{-2,8\}

:

(A)increasing in $(-2,8)$ and decreasing in $(-\infty,-2)\cup(8,\infty)$
(B)decreasing in $(-\infty,-2)\cup(-2,8)$ and increasing in $(8,\infty)$
(C)decreasing in $(-\infty,-2)\cup(8,\infty)$ and increasing in $(-2,8)$
(D)decreasing in $\mathbb{R}-\{-2,8\}$

Q77·MathematicsSingle correctJEE Main 2024

The function

f(x)=2x+3(x)^{\frac{2}{3}},\ x\in\mathbb{R}

, has:

(A)exactly one point of local minima and no point of local maxima
(B)exactly one point of local maxima and no point of local minima
(C)exactly one point of local maxima and exactly one point of local minima
(D)exactly two points of local maxima and exactly one point of local minima

Q78·MathematicsNumericalJEE Main 2024

Let

f(x)=\sqrt{\displaystyle\lim_{r\to x}\left\{\dfrac{2r^2[(f(r))^2-f(x)f(r)]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}

be differentiable in

(-\infty,0)\cup(0,\infty)

and

f(1)=1

. Then the value of

ea

, such that

f(a)=0

, is equal to ___.

Q79·MathematicsSingle correctJEE Main 2024

Let

y=\log_e\left(\dfrac{1-x^2}{1+x^2}\right),\ -1<x<1

. Then at

x=\dfrac{1}{2}

, the value of

225(y'-y'')

is equal to:

(A)732
(B)746
(C)742
(D)736

Q80·MathematicsSingle correctJEE Main 2024

\displaystyle\lim_{x\to\frac{\pi}{2}}\left(\dfrac{1}{\left(x-\dfrac{\pi}{2}\right)^2}\int_{x^3}^{\left(\frac{\pi}{2}\right)^3}\cos\left(t^{1/3}\right)dt\right)

is equal to

(A) $\dfrac{3\pi}{8}$
(B) $\dfrac{3\pi^2}{4}$
(C) $\dfrac{3\pi^2}{8}$
(D) $\dfrac{3\pi}{4}$

Q81·MathematicsNumericalJEE Main 2024

Let the slope of the line

45x+5y+3=0

be

27r_1+\dfrac{9r_2}{2}

for some

r_1,r_2\in\mathbb{R}

. Then

\displaystyle\lim_{x\to 3}\left(\int_3^x\dfrac{8t^2}{\dfrac{3r_2 x}{2}-r_2 x^2-r_1 x^3-3x}\,dt\right)

is equal to ___.

Q82·MathematicsSingle correctJEE Main 2024

Suppose

f(x)=\dfrac{\left(2^x+2^{-x}\right)\tan x\sqrt{\tan^{-1}\left(x^2-x+1\right)}}{\left(7x^2+3x+1\right)^3}

. Then the value of

f'(0)

is equal to

(A) $\pi$
(B) $0$
(C) $\sqrt{\pi}$
(D) $\dfrac{\pi}{2}$

Q83·MathematicsSingle correctJEE Main 2024

Consider the function

f:\left[\dfrac{1}{2},1\right]\to R

defined by

f(x)=4\sqrt{2}x^3-3\sqrt{2}x-1

. Consider the statements (I) The curve

y=f(x)

intersects the x-axis exactly at one point (II) The curve

y=f(x)

intersects the x-axis at

x=\cos\dfrac{\pi}{12}

. Then

(A)Only (II) is correct
(B)Both (I) and (II) are incorrect
(C)Only (I) is correct
(D)Both (I) and (II) are correct

Q84·MathematicsSingle correctJEE Main 2024

If

\displaystyle\lim_{x\to 0}\dfrac{3+\alpha\sin x+\beta\cos x+\log_e(1-x)}{3\tan^2 x}=\dfrac13

, then

2\alpha-\beta

is equal to :

(A)2
(B)7
(C)5
(D)1

Q85·MathematicsNumericalJEE Main 2024

Let

f(x)=x^3+x^2 f'(1)+x\,f''(2)+f'''(3)

,

x\in\mathbb R

. Then

f'(10)

is equal to __________.

Q86·MathematicsSingle correctJEE Main 2024

Consider the function

f:(0,2)\to\mathbb{R}

defined by

f(x)=\dfrac{x}{2}+\dfrac{2}{x}

and the function

g(x)

defined by

g(x)=\begin{cases}\min\{f(t)\},\ 0<t\le x\text{ and }0<x\le 1 \\ \dfrac{3}{2}+x,\ 1<x<2\end{cases}

. Then

(A) $g$ is continuous but not differentiable at $x=1$
(B) $g$ is not continuous for all $x\in(0,2)$
(C) $g$ is neither continuous nor differentiable at $x=1$
(D) $g$ is continuous and differentiable for all $x\in(0,2)$

Q87·MathematicsSingle correctJEE Main 2024

If

a=\displaystyle\lim_{x\to 0}\dfrac{\sqrt{1+\sqrt{1+x^4}}-\sqrt2}{x^4}

and

b=\displaystyle\lim_{x\to 0}\dfrac{\sin^2 x}{\sqrt2-\sqrt{1+\cos x}}

, then the value of

ab^3

is:

(A)36
(B)32
(C)25
(D)30

Q88·MathematicsSingle correctJEE Main 2024

Consider the function

f(x)=\begin{cases}\dfrac{a(7x-12-x^2)}{b\,|x^2-7x+12|}, & x<3\\ 2^{\sin(x-3)/(x-[x])}, & x>3\\ b, & x=3\end{cases}

, where

[x]

denotes the greatest integer less than or equal to

x

. If

S

denotes the set of all ordered pairs

(a,b)

such that

f(x)

is continuous at

x=3

, then the number of elements in

S

is:

(A)2
(B)Infinitely many
(C)4
(D)1

Q89·MathematicsSingle correctJEE Advanced 2023

Let

f : (0, 1) \rightarrow R

be the function defined as

f(x) = \sqrt{n}

if

x \in \left[\frac{1}{n+1}, \frac{1}{n}\right)

where

n \in N

. Let

g : (0, 1) \rightarrow R

be a function such that

\int_{x^2}^{x} \sqrt{\frac{1-t}{t}}\,dt < g(x) < 2\sqrt{x}

for all

x \in (0, 1)

. Then

\lim_{x \rightarrow 0} f(x)g(x)

(A)does NOT exist
(B)is equal to 1
(C)is equal to 2
(D)is equal to 3

Q90·MathematicsSingle correctJEE Main 2023

If

\lim\limits_{x \to 0} \dfrac{e^{ax} - \cos(bx) - \dfrac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17

, then

5a^2 + b^2

is

(A)72
(B)76
(C)68
(D)64

Q91·MathematicsNumericalJEE Main 2023

Let

[x]

be the greatest integer

\le x

. Then the number of points in the interval

(-2,1)

, where the function

f(x)=|[x]|+\sqrt{x-[x]}

is discontinuous is _________.

Q92·MathematicsSingle correctJEE Main 2023

Let

g

and

f

be two functions defined as

f(x)=\begin{cases} x+1, & x<0 \\ |x-1|, & x\ge 0 \end{cases}

and

g(x)=\begin{cases} x+1, & x<0 \\ 1, & x\ge 0 \end{cases}

. Then

(g\circ f)(x)

is

(A)differentiable everywhere
(B)continuous everywhere but not differentiable exactly at one point
(C)not continuous at $x=-1$
(D)continuous everywhere but not differentiable at $x=1$

Q93·MathematicsNumericalJEE Main 2023

Let

f:(-2,2)\to\mathbb{R}

be defined by

f(x)=\begin{cases}x[x] & ,-2<x<0\\(x-1)[x] & ,0\le x<2\end{cases}

where

[x]

denotes the greatest integer function. If m and n respectively are the number of points in

(-2,2)

at which

y=|f(x)|

is not continuous and not differentiable, then

m+n

is equal to _________.

Q94·MathematicsSingle correctJEE Main 2023

If

\alpha>\beta>0

are the roots of the equation

ax^{2}+bx+1=0

, and

\displaystyle\lim_{x\to\frac{1}{\alpha}}\left(\dfrac{1-\cos(x^{2}+bx+a)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\dfrac{1}{k}\left(\dfrac{1}{\beta}-\dfrac{1}{\alpha}\right)

, then k is equal to

(A) $2\beta$
(B) $2\alpha$
(C) $\alpha$
(D) $\beta$

Q95·MathematicsSingle correctJEE Main 2023

\lim_{x\to 0}\left(\left(\frac{1-\cos^{2}(3x)}{\cos^{3}(4x)}\right)\left(\frac{\sin^{3}(4x)}{(\log_{e}(2x+1))^{5}}\right)\right)

is equal to

(A)9
(B)18
(C)15
(D)24

Q96·MathematicsSingle correctJEE Main 2023

\lim_{n\to\infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right)\cdots\left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\right)\right\}

is equal to

(A) $\frac{1}{\sqrt{2}}$
(B)1
(C) $\sqrt{2}$
(D)0

Q97·MathematicsNumericalJEE Main 2023

Let

a\in\mathbb{Z}

and

[t]

be the greatest integer

\le t

. Then the number of points, where the function

f(x)=[a+13\sin x],\,x\in(0,\pi)

is not differentiable, is _____.

Q98·MathematicsSingle correctJEE Main 2023

\displaystyle\lim_{x\to\infty}\dfrac{(\sqrt{3x+1}+\sqrt{3x-1})^6+(\sqrt{3x+1}-\sqrt{3x-1})^6}{(x+\sqrt{x^2-1})^6+(x-\sqrt{x^2-1})^6}x^3

(A)does not exist
(B)is equal to $27$
(C)is equal to $\dfrac{27}{2}$
(D)is equal to $9$

Q99·MathematicsSingle correctJEE Main 2023

Let

f, g

and

h

be the real valued functions defined on

\mathbb{R}

as

f(x)=\begin{cases} \dfrac{x}{|x|}, & x \ne 0 \\ 1, & x=0 \end{cases}

,

g(x)=\begin{cases} \dfrac{\sin(x + 1)}{(x + 1)}, & x \ne -1 \\ 1, & x=-1 \end{cases}

and

h(x)=2[x] - f(x)

, where

[x]

is the greatest integer

\le x

. Then the value of

\displaystyle\lim_{x \to 1} g(h(x - 1))

is:

(A) $-1$
(B) $0$
(C) $\sin(1)$
(D) $1$

Q100·MathematicsNumericalJEE Main 2023

\lim_{x\to0}\dfrac{48}{x^4}\int_0^x\dfrac{t^3}{t^4+1}\,dt

is equal to

Q101·MathematicsSingle correctJEE Main 2023

Let

x = 2

be a root of the equation

x^2 + px + q = 0

and

f(x) = \begin{cases} \dfrac{1 - \cos(x^2 - 4px + q^2 + 8q + 16)}{(x - 2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}

. Then

\lim_{x \to 2p^+} [f(x)]

, where

[\cdot]

denotes greatest integer function, is

(A)0
(B) $-1$
(C)2
(D)1

Q102·MathematicsSingle correctJEE Main 2023

If the function

f(x) = \begin{cases} (1 + |\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0 < x < \dfrac{\pi}{2} \\ \mu, & x = \dfrac{\pi}{2} \\ \dfrac{\cot 6x}{e^{\cot 4x}}, & \dfrac{\pi}{2} < x < \pi \end{cases}

is continuous at

x = \dfrac{\pi}{2}

, then

9\lambda + 6\log_e \mu + \mu^6 - e^{6\lambda}

is equal to

(A)10
(B) $2e^4 + 8$
(C)11
(D)8

Q103·MathematicsSingle correctJEE Main 2023

The value of

\displaystyle\lim_{n\to\infty}\dfrac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}

is:

(A) $\dfrac{3}{2}(\sqrt{2}+1)$
(B) $\dfrac{3}{2\sqrt{2}}$
(C) $\dfrac{\sqrt{2}+1}{2}$
(D) $3(\sqrt{2}+1)$

Q104·MathematicsSingle correctJEE Main 2023

The set of all values of

a

for which

\lim_{x\to a}([x-5]-[2x+2])=0

, where

[\alpha]

denotes the greatest integer less than or equal to

\alpha

, is equal to

(A) $[-7.5,-6.5)$
(B) $[-7.5,-6.5]$
(C) $(-7.5,-6.5]$
(D) $(-7.5,-6.5)$

Q105·MathematicsSingle correctJEE Main 2023

\lim\limits_{t\to 0}\left(1^{\frac{1}{\sin^2 t}}+2^{\frac{1}{\sin^2 t}}+\cdots+n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}

is equal to

(A) $n^2$
(B) $\tfrac{n(n+1)}{2}$
(C) $n$
(D) $n^2+n$

Q106·MathematicsNumericalJEE Advanced 2022

Let

\alpha

be a positive real number. Let

f : R \rightarrow R

and

g : (\alpha, \infty) \rightarrow R

be the functions defined by

f(x) = \sin\left(\frac{\pi x}{12}\right)

and

g(x) = \frac{2\log_{e}\left(\sqrt{x} - \sqrt{\alpha}\right)}{\log_{e}\left(e^{\sqrt{x}} - e^{\sqrt{\alpha}}\right)}

. Then the value of

\lim_{x \rightarrow \alpha^{+}} f\left(g(x)\right)

is __________.

Q107·MathematicsIntegerJEE Advanced 2022

If

\beta = \lim_{x \to 0} \frac{e^{x^{3}} - \left(1 - x^{3}\right)^{1/3} + \left(\left(1 - x^{2}\right)^{1/2} - 1\right)\sin x}{x\sin^{2}x}

, then the value of 6β is ________.

Q108·MathematicsSingle correctJEE Main 2022

Let the function

f(x) = \begin{cases} \dfrac{\log_e (1 + 5x) - \log_e (1 + \alpha x)}{x} & ; \text{if } x \neq 0 \\ 10 & ; \text{if } x = 0 \end{cases}

be continuous at

x = 0

. The

\alpha

is equal to :

(A) $10$
(B) $-10$
(C) $5$
(D) $-5$

Q109·MathematicsSingle correctJEE Main 2022

If

\lim_{x \to 0} \frac{\alpha e^{x} + \beta e^{-x} + \gamma \sin x}{x \sin^{2} x} = \frac{2}{3}

, where

\alpha, \beta, \gamma \in R

, then which of the following is NOT correct ?

(A) $\alpha^{2} + \beta^{2} + \gamma^{2} = 6$
(B) $\alpha\beta + \beta\gamma + \gamma\alpha + 1 = 0$
(C) $\alpha\beta^{2} + \beta\gamma^{2} + \gamma\alpha^{2} + 3 = 0$
(D) $\alpha^{2} - \beta^{2} + \gamma^{2} = 4$

Q110·MathematicsSingle correctJEE Main 2022

The function

f: R \rightarrow R

defined by

f(x)=\lim_{n \rightarrow \infty} \frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}}

is continuous for all x in

(A) $R-\{-1\}$
(B) $R-\{-1,1\}$
(C) $R-\{1\}$
(D) $R-\{0\}$

Q111·MathematicsNumericalJEE Main 2022

\lim_{x \to 0} \left( \frac{\left(x + 2\cos x\right)^{3} + 2\left(x + 2\cos x\right)^{2} + 3\sin\left(x + 2\cos x\right)}{\left(x + 2\right)^{3} + 2\left(x + 2\right)^{2} + 3\sin\left(x + 2\right)} \right)^{\frac{100}{x}}

is equal to ___________.

Q112·MathematicsSingle correctJEE Main 2022

Let a function

f:\mathbb{R}\rightarrow\mathbb{R}

be defined as :

f\left(x\right)=\begin{cases}\int_{0}^{x}\left(5-\left|t-3\right|\right)dt, & x>4\\ x^{2}+bx, & x\le 4\end{cases}

where

b\in\mathbb{R}

. If

f

is continuous at x = 4, then which of the following statements is NOT true ?

(A) $f$ is not differentiable at x = 4
(B) $f\,'\left(3\right)+f\,'\left(5\right)=\frac{35}{4}$
(C) $f$ is increasing in $\left(-\infty,\frac{1}{8}\right)\cup\left(8,\infty\right)$
(D) $f$ has a local minima at $x=\frac{1}{8}$

Q113·MathematicsSingle correctJEE Main 2022

If for

p \neq q \neq 0

, then function

f(x) = \frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}

is continuous at

x = 0

, then:

(A) $7pq\,f(0)-1=0$
(B) $63q\,f(0)-p^{2}=0$
(C) $21q\,f(0)-p^{2}=0$
(D) $7pq\,f(0)-9=0$

Q114·MathematicsSingle correctJEE Main 2022

If

f(x) = \begin{cases} x + a, & x \le 0 \\ |x - 4|, & x > 0 \end{cases}

and

g(x) = \begin{cases} x + 1, & x < 0 \\ (x - 4)^2 + b, & x \ge 0 \end{cases}

are continuous on R, then (gof) (2) + (fog) (−2) is equal to :

(A)−10
(B)10
(C)8
(D)−8

Q115·MathematicsSingle correctJEE Main 2022

If the function

f(x) = \begin{cases} \dfrac{\log_e(1 - x + x^2) + \log_e(1 + x + x^2)}{\sec x - \cos x}, & x \in \left( \dfrac{-\pi}{2}, \dfrac{\pi}{2} \right) - \{0\} \\ k, & x = 0 \end{cases}

is continuous at x = 0, then k is equal to :

(A)1
(B)−1
(C) $e$
(D)0

Q116·MathematicsSingle correctJEE Main 2022

Let

\beta=\lim\limits_{x\to 0}\frac{\alpha x-\left(e^{3x}-1\right)}{\alpha x\left(e^{3x}-1\right)}

for some

\alpha\in\mathbb{R}

. Then the value of

\alpha+\beta

is :

(A) $\frac{14}{5}$
(B) $\frac{3}{2}$
(C) $\frac{5}{2}$
(D) $\frac{7}{2}$

Q117·MathematicsSingle correctJEE Main 2022

If

\lim_{n \to \infty} \left( \sqrt{n^{2} - n - 1} + n\alpha + \beta \right) = 0

then

8(\alpha + \beta)

is equal to :

(A) $4$
(B) $-8$
(C) $-4$
(D) $8$

Q118·MathematicsNumericalJEE Main 2022

Let

f(x) = \begin{cases} |4x^{2} - 8x + 5|, & \text{if } 8x^{2} - 6x + 1 \ge 0 \\ [4x^{2} - 8x + 5], & \text{if } 8x^{2} - 6x + 1 < 0 \end{cases}

, where

[\alpha]

denotes the greatest integer less than or equal to

\alpha

. Then the number of points in R where f is not differentiable is ________.

Q119·MathematicsSingle correctJEE Main 2022

The value of

\lim_{x \to 1} \frac{\left(x^{2} - 1\right)\sin^{2}\left(\pi x\right)}{x^{4} - 2x^{3} + 2x - 1}

is equal to:

(A) $\frac{\pi^{2}}{6}$
(B) $\frac{\pi^{2}}{3}$
(C) $\frac{\pi^{2}}{2}$
(D) $\pi^{2}$

Q120·MathematicsSingle correctJEE Main 2022

Let

f, g : \mathbb{R} \to \mathbb{R}

be functions defined by

f(x) = \begin{cases} [x] & , \quad x < 0 \\ |1 - x| & , \quad x \geq 0 \end{cases}

and

g(x) = \begin{cases} e^{x} - x & , \quad x < 0 \\ (x-1)^{2} - 1 & , \quad x \geq 0 \end{cases}

where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :

(A)one point
(B)two points
(C)three points
(D)four points

Q121·MathematicsNumericalJEE Main 2022

If

\lim\limits_{x\to 1} \frac{\sin\left(3x^2 - 4x + 1\right) - x^2 + 1}{2x^3 - 7x^2 + ax + b} = -2

, then the value of

(a - b)

is equal to

Q122·MathematicsSingle correctJEE Main 2022

Let

\mathrm{f}:\mathbb{R} \to \mathbb{R}

be defined as

\mathrm{f(x)}=\begin{cases} [e^{x}], & x<0 \\ ae^{x}+[x-1], & 0 \le x < 1 \\ b+[\sin(\pi x)], & 1 \le x < 2 \\ [e^{-x}]-c, & x \ge 2 \end{cases}

where

a, b, c \in \mathbb{R}

and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true ?

(A)There exists $a, b, c \in \mathbb{R}$ such that f is continuous of $\mathbb{R}$ .
(B)If f is discontinuous at exactly one point, then $a + b + c = 1$ .
(C)If f is discontinuous at exactly one point, then $a + b + c \ne 1$ .
(D)f is discontinuous at atleast two points, for any values of a, b and c.

Q123·MathematicsSingle correctJEE Main 2022

Let a be an integer such that

\lim_{x \to 7} \frac{18 - \left[1 - x\right]}{\left[x - 3a\right]}

exists, where [t] is greatest integer

\leq

t. Then a is equal to :

(A)-6
(B)-2
(C)2
(D)6

Q124·MathematicsNumericalJEE Main 2022

Let [t] denote the greatest integer

\le

t and {t} denote the fractional part of t. Then integral value of

\alpha

for which the left hand limit of the function

f(x)=[1+x]+\dfrac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}

at x = 0 is equal to

\alpha-\dfrac{4}{3}

is ______

Q125·MathematicsSingle correctJEE Main 2022

Let f (x) = min {1, 1 + x sin x}, 0 ≤ x ≤ 2π. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

(A)(2, 0)
(B)(1, 0)
(C)(1, 1)
(D)(2, 1)

Q126·MathematicsSingle correctJEE Main 2022

\lim_{x\to 0}\frac{\cos(\sin x)-\cos x}{x^4}

is equal to :

(A) $\frac{1}{3}$
(B) $\frac{1}{4}$
(C) $\frac{1}{6}$
(D) $\frac{1}{12}$

Q127·MathematicsNumericalJEE Main 2022

Let

f:\mathbb{R}\to\mathbb{R}

satisfy

f(x+y)=2^{x}f(y)+4^{y}f(x)

, ∀x, y ∈

\mathbb{R}

. If

f(2)=3

, then

14\cdot\frac{f'(4)}{f'(2)}

is equal to ______.

Q128·MathematicsSingle correctJEE Main 2022

Let f, g : R

\to

R be two real valued functions defined as

f(x) = \begin{cases} -|x+3| & , & x < 0 \\ e^{x} & , & x \geq 0 \end{cases}

and

g(x) = \begin{cases} x^{2} + k_{1}x & , & x < 0 \\ 4x + k_{2} & , & x \geq 0 \end{cases}

, where

k_{1}

and

k_{2}

are real constants. If (gof) is differentiable at x = 0, then (gof) (-4) + (gof) (4) is equal to :

(A) $4(e^{4} + 1)$
(B) $2(2e^{4} + 1)$
(C) $4e^{4}$
(D) $2(2e^{4} - 1)$

Q129·MathematicsSingle correctJEE Main 2022

\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)}

is equal to :

(A) $\sqrt{2}$
(B) $-\sqrt{2}$
(C) $\frac{1}{\sqrt{2}}$
(D) $-\frac{1}{\sqrt{2}}$

Q130·MathematicsSingle correctJEE Main 2022

Let

f(x)

be a polynomial function such that

f(x) + f'(x) + f''(x) = x^5 + 64

. Then, the value of

\lim_{x \to 1} \frac{f(x)}{x - 1}

(A) $-15$
(B) $-60$
(C) $60$
(D) $15$

Q131·MathematicsNumericalJEE Main 2022

Let

f(x) = [2x^{2} + 1]

and

g(x) = \begin{cases} 2x - 3, & x < 0 \\ 2x + 3, & x \geq 0 \end{cases}

, where [t] is the greatest integer

\leq t

. Then, in the open interval

(-1, 1)

, the number of points where fog is discontinuous is equal to ______

Q132·MathematicsSingle correctJEE Main 2022

\lim_{x \to \frac{\pi}{2}}\left( \tan^2 x \left( \left(2\sin^2 x + 3\sin x + 4\right)^{\frac{1}{2}} - \left(\sin^2 x + 6\sin x + 2\right)^{\frac{1}{2}} \right) \right)

is equal to

(A) $\frac{1}{12}$
(B) $-\frac{1}{18}$
(C) $-\frac{1}{12}$
(D) $-\frac{1}{6}$

Q133·MathematicsNumericalJEE Main 2022

The number of points where the function

f(x)=\begin{cases}|2x^{2}-3x-7| & \text{if } x\le -1\\ [4x^{2}-1] & \text{if } -1<x<1\\ |x+1|+|x-2| & \text{if } x\ge 1\end{cases}

[t]

denotes the greatest integer

\le t

, is discontinuous is ______ .

Q134·MathematicsSingle correctJEE Main 2022

Let

f(x) = \begin{cases} \dfrac{\sin\left(x-[x]\right)}{x-[x]} &,& x \in (-2,-1) \\ \max\left\{2x, 3\left[|x|\right]\right\} &,& |x| < 1 \\ 1 &,& \text{otherwise} \end{cases}

where

[t]

denotes greatest integer

\le t

. If m is the number of points where

f

is not continuous and n is the number of points where

f

is not differentiable, then the ordered pair (m, n) is :

(A)(3, 3)
(B)(2, 4)
(C)(2, 3)
(D)(3, 4)

Q135·MathematicsNumericalJEE Main 2021

Let

f(x) = x^{6} + 2x^{4} + x^{3} + 2x + 3

, x ∈ ℝ. Then the natural number n for which

\lim_{x \to 1} \frac{x^{n} f(1) - f(x)}{x - 1} = 44

is ________ .

Q136·MathematicsSingle correctJEE Main 2021

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous function. Then

\lim_{x \to \frac{\pi}{4}} \frac{\frac{\pi}{4}\int_{2}^{\sec^{2} x} f(x)\,dx}{x^{2} - \frac{\pi^{2}}{16}}

is equal to :

(A) $f(2)$
(B) $2f(2)$
(C) $2f\left(\sqrt{2}\right)$
(D) $4f(2)$

Q137·MathematicsNumericalJEE Main 2021

Let [t] denote the greatest integer ≤ t. The number of points where the function

f(x) = [x]\left|x^{2} - 1\right| + \sin\left(\frac{\pi}{[x] + 3}\right) - [x + 1], x \in (-2, 2)

is not continuous is ______ .

Q138·MathematicsSingle correctJEE Main 2021

\lim_{x \to 0} \frac{\sin^{2}\left(\pi \cos^{4} x\right)}{x^{4}}

is equal to :

(A) $\pi^{2}$
(B) $2\pi^{2}$
(C) $4\pi^{2}$
(D) $4\pi$

Q139·MathematicsSingle correctJEE Main 2021

If

\alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}

and

\beta = \lim_{x \to 0} (\cos x)^{\cot x}

are the roots of the equation,

ax^2 + bx - 4 = 0

, then the ordered pair

(a, b)

is :

(A) $(1, -3)$
(B) $(-1, 3)$
(C) $(-1, -3)$
(D) $(1, 3)$

Q140·MathematicsSingle correctJEE Main 2021

If the function

f(x) = \begin{cases} \frac{1}{x} \log_{e}\left(\frac{1 + \frac{x}{a}}{1 - \frac{x}{b}}\right), & x < 0 \\ k, & x = 0 \\ \frac{\cos^{2} x - \sin^{2} x - 1}{\sqrt{x^{2} + 1} - 1}, & x > 0 \end{cases}

is continuous at

x = 0

, then

\frac{1}{a} + \frac{1}{b} + \frac{4}{k}

is equal to :

(A) $-5$
(B)5
(C) $-4$
(D)4

Q141·MathematicsSingle correctJEE Main 2021

If

\alpha, \beta

are the distinct roots of

x^2 + bx + c = 0

, then

\lim_{x \to \beta} \frac{e^{2\left(x^2+bx+c\right)} - 1 - 2\left(x^2 + bx + c\right)}{\left(x - \beta\right)^2}

is equal to:

(A) $b^2 + 4c$
(B) $2(b^2 + 4c)$
(C) $2(b^2 - 4c)$
(D) $b^2 - 4c$

Q142·MathematicsSingle correctJEE Main 2021

If

\lim_{x \to \infty}\left(\sqrt{x^{2} - x + 1} - ax\right) = b

, then the ordered pair (a, b) is:

(A) $\left(1, \frac{1}{2}\right)$
(B) $\left(1, -\frac{1}{2}\right)$
(C) $\left(-1, \frac{1}{2}\right)$
(D) $\left(-1, -\frac{1}{2}\right)$

Q143·MathematicsSingle correctJEE Main 2021

If

U_n = \left(1 + \frac{1}{n^2}\right)\left(1 + \frac{2^2}{n^2}\right)^2 .... \left(1 + \frac{n^2}{n^2}\right)^n

, then

\lim_{n \to \infty} (U_n)^{\frac{-4}{n^2}}

is equal to :

(A) $\frac{e^2}{16}$
(B) $\frac{4}{e}$
(C) $\frac{16}{e^2}$
(D) $\frac{4}{e^2}$

Q144·MathematicsNumericalJEE Main 2021

Let a, b

\in

R,

b \neq 0

, Define a function

f(x) = \begin{cases} a\sin\frac{\pi}{2}(x-1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^{3}}, & \text{for } x > 0. \end{cases}

If f is continuous at

x = 0

, then

10 - ab

is equal to ________.

Q145·MathematicsSingle correctJEE Main 2021

\lim_{x \to 2}\left(\sum_{n=1}^{9} \frac{x}{n(n + 1)x^{2} + 2(2n + 1)x + 4}\right)

is equal to :

(A) $\frac{9}{44}$
(B) $\frac{5}{24}$
(C) $\frac{1}{5}$
(D) $\frac{7}{36}$

Q146·MathematicsSingle correctJEE Main 2021

The value of

\lim_{x \to 0} \left( \frac{x}{\sqrt[8]{1 - \sin x} - \sqrt[8]{1 + \sin x}} \right)

is equal to :

(A)4
(B)0
(C)−4
(D)−1

Q147·MathematicsSingle correctJEE Main 2021

Let

f:R \to R

be a function such that

f\left(2\right)=4

and

f'\left(2\right)=1

. Then the value of

\lim_{x \to 2}\frac{x^2 f\left(2\right)-4f\left(x\right)}{x-2}

is equal to :

(A)12
(B)4
(C)16
(D)8

Q148·MathematicsNumericalJEE Main 2021

Let

f:\left[0,3\right] \to R

be defined by

f\left(x\right)=\min\left\{x-\left[x\right],1+\left[x\right]-x\right\}

Where

\left[x\right]

is the greatest integer less than or equal to x. Let P denote the set containing all

x \in \left[0,3\right]

where f is discontinuous, and Q denote the set containing all

x \in \left(0,3\right)

where f is not differentiable. Then the sum of number of elements in P and Q is equal to ____.

Q149·MathematicsSingle correctJEE Main 2021

Let

f : \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \to R

be defined as

f(x) = \begin{cases} \left(1 + |\sin x|\right)^{\frac{3a}{|\sin x|}} & , \ -\frac{\pi}{4} < x < 0 \\ b & , \ x = 0 \\ e^{\cot 4x / \cot 2x} & , \ 0 < x < \frac{\pi}{4} \end{cases}

If f is continuous at x = 0, then the value of

6a + b^2

is equal to :

(A) $1 - e$
(B)e
(C) $1 + e$
(D) $e - 1$

Q150·MathematicsSingle correctJEE Main 2021

Let f : R → R be defined as

f(x) = \begin{cases} \dfrac{\lambda\left|x^{2} - 5x + 6\right|}{\mu\left(5x - x^{2} - 6\right)}, & x < 2 \\ e^{\frac{\tan(x-2)}{x-[x]}}, & x > 2 \\ \mu, & x = 2 \end{cases}

where [x] is the greatest integer less than or equal to x. If f is continuous at x = 2, then

\lambda + \mu

is equal to :

(A) $e(-e + 1)$
(B) $e(e - 2)$
(C) $2e - 1$
(D)1

Q151·MathematicsSingle correctJEE Main 2021

If [x] be the greatest integer less than or equal to x, then

\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]

is equal to :

(A)4
(B)-2
(C)2
(D)0

Q152·MathematicsNumericalJEE Main 2021

Consider the function

f(x) = \dfrac{P(x)}{\sin(x-2)}

,

x \neq 2

= 7

,

x = 2

where P(x) is a polynomial such that P''(x) is always a constant and P(3) = 9. If (x) is continuous at x = 2, then P(5) is equal to.......

Q153·MathematicsSingle correctJEE Main 2021

The lowest integer which is greater than

\left(1 + \frac{1}{10^{100}}\right)^{10^{100}}

is..........

(A)3
(B)2
(C)4
(D)1

Q154·MathematicsSingle correctJEE Main 2021

Let

f : R \rightarrow R

be defined as

f(x) = \begin{cases} \dfrac{x^{3}}{(1-\cos 2x)^{2}} \log_{e}\left(\dfrac{1+2xe^{-2x}}{\left(1-xe^{-x}\right)^{2}}\right) , & x \neq 0 \\ \alpha , & x = 0 \end{cases}

. If

f

is continuous at x = 0, then

\alpha

is equal to :

(A)0
(B)1
(C)2
(D)3

Q155·MathematicsNumericalJEE Main 2021

If the value of

\lim_{x \to 0} \left(2 - \cos x \sqrt{\cos 2x}\right)^{\left(\frac{x + 2}{x^2}\right)}

is equal to

e^a

, then a is equal to.......

Q156·MathematicsSingle correctJEE Main 2021

Let a function f : R → R be defined as

f(x) = \begin{cases} \sin x - e^x & \text{if } x \leq 0 \\ a + [-x] & \text{if } 0 < x < 1 \\ 2x - b & \text{if } x \geq 1 \end{cases}

where [x] is the greatest integer less than or equal to x. If f is continuous on R, then

\left(a+b\right)

is equal to :

(A)2
(B)5
(C)3
(D)4

Q157·MathematicsSingle correctJEE Main 2021

Let f : R → R be a function defined as

f(x) = \begin{cases} \dfrac{\sin(a+1)x + \sin 2x}{2x} & ,\ \text{if } x < 0 \\ b & ,\ \text{if } x = 0 \\ \dfrac{\sqrt{x + bx^{3}} - \sqrt{x}}{bx^{5/2}} & ,\ \text{if } x > 0 \end{cases}

If f is continuous at x = 0, then the value of a + b is equal to :

(A) $-\frac{5}{2}$
(B)−2
(C)−3
(D) $-\frac{3}{2}$

Q158·MathematicsSingle correctJEE Main 2021

If

\lim_{x \to 0} \dfrac{\sin^{-1}x - \tan^{-1}x}{3x^{3}}

is equal to L, then the value of

(6L + 1)

is

(A) $\dfrac{1}{6}$
(B) $\dfrac{1}{2}$
(C) $6$
(D) $2$

Q159·MathematicsNumericalJEE Main 2021

If the function

f(x) = \frac{\cos(\sin x) - \cos x}{x^4}

is continuous at each point in its domain and

f(0) = \frac{1}{k}

, then k is _________ .

Q160·MathematicsSingle correctJEE Main 2021

The value of

\lim_{n\rightarrow\infty}\frac{[r]+[2r]+.....+[nr]}{n^{2}}

, where

r

is non-zero real number and

[r]

denotes the greatest integer less than or equal to

r

, is equal to :

(A) $\frac{r}{2}$
(B) $r$
(C) $2r$
(D) $0$

Q161·MathematicsSingle correctJEE Main 2021

The value of the limit

\lim_{\theta\rightarrow 0}\frac{\tan\left(\pi\cos^{2}\theta\right)}{\sin\left(2\pi\sin^{2}\theta\right)}

is equal to :

(A) $-\frac{1}{2}$
(B) $-\frac{1}{4}$
(C) $0$
(D) $\frac{1}{4}$

Q162·MathematicsSingle correctJEE Main 2021

The value of

\lim_{x \to 0^{+}} \frac{\cos^{-1}\left(x - [x]^2\right) \cdot \sin^{-1}\left(x - [x]^2\right)}{x - x^3}

, where

[x]

denotes the greatest integer

\leq x

is :

(A) $\pi$
(B)0
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{2}$

Q163·MathematicsNumericalJEE Main 2021

Let

f : R \rightarrow R

and

g : R \rightarrow R

be defined as

f(x)=\begin{cases} x+a, & x<0 \\ |x-1|, & x \ge 0 \end{cases}

and

g(x)=\begin{cases} x+1, & x<0 \\ (x-1)^{2}+b, & x \ge 0 \end{cases}

where a, b are non-negative real numbers. If

(gof)(x)

is continuous for all

x \in R

, then

a + b

is equal to ___________.

Q164·MathematicsNumericalJEE Main 2021

If

\lim_{x \to 0}\frac{ae^{x} - b\cos x + ce^{-x}}{x\sin x} = 2

, then a + b + c is equal to _______.

Q165·MathematicsSingle correctJEE Main 2021

Let

\alpha \in R

be such that the function

f(x)=\begin{cases} \dfrac{\cos^{-1}(1-\{x\}^{2})\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^{3}}, & x \neq 0 \\ \alpha, & x=0 \end{cases}

is continuous at

x = 0

, where

\{x\} = x - [x]

,

[x]

is the greatest integer less than or equal to x. Then :

(A) $\alpha = \frac{\pi}{\sqrt{2}}$
(B) $\alpha = 0$
(C)no such $\alpha$ exists
(D) $\alpha = \frac{\pi}{4}$

Q166·MathematicsSingle correctJEE Main 2021

Let f(x) be a differentiable function at x = a with

f'(a) = 2

and f(a) = 4. Then

\lim_{x \to a} \frac{xf(a) - af(x)}{x - a}

equals :

(A)2a + 4
(B)2a − 4
(C)4 − 2a
(D)a + 4

Q167·MathematicsSingle correctJEE Main 2021

Let f : R → R be defined as

f(x) = \begin{cases} 2\sin\left(-\frac{\pi x}{2}\right), & \text{if } x < -1 \\ |ax^{2} + x + b|, & \text{if } -1 \le x \le 1 \\ \sin(\pi x) & \text{if } x > 1 \end{cases}

If f(x) is continuous on R, then a + b equals :

(A)3
(B)−1
(C)−3
(D)1

Q168·MathematicsSingle correctJEE Main 2021

The value of

\lim_{h \to 0}\left\{\frac{\sqrt{3}\,\sin\left(\frac{\pi}{6}+h\right)-\cos\left(\frac{\pi}{6}+h\right)}{\sqrt{3}h\left(\sqrt{3}\,\cos h-\sin h\right)}\right\}

is:

(A) $\frac{3}{4}$
(B) $\frac{2}{\sqrt{3}}$
(C) $\frac{4}{3}$
(D) $\frac{2}{3}$

Q169·MathematicsSingle correctJEE Main 2021

\lim\limits_{n \to \infty} \left( 1 + \dfrac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{n^2} \right)^{n}

is equal to :

(A) $\frac{1}{2}$
(B) $\frac{1}{e}$
(C) $1$
(D) $0$

Q170·MathematicsNumericalJEE Main 2021

If

_{x}

lim

_{→}_{0}

axax− (e(e

_{4x}^{4x}

−−11)) exists and is equal to b, then the value of a – 2b is ______.

Q171·MathematicsSingle correctJEE Main 2021

If

f : R \rightarrow R

is a function defined by

f(x) = [x-1] \cos\left(\frac{2x - 1}{2}\right)\pi

, where

[.]

denotes the greatest integer function, then

f

is :

(A)discontinuous only at $x = 1$
(B)discontinuous at all integral values of $x$ except at $x = 1$
(C)continuous only at $x = 1$
(D)continuous for every real $x$

Q172·MathematicsNumericalJEE Advanced 2020

Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

\lim_{x \to 0^{+}} \frac{(1 - x)^{\frac{1}{x}} - e^{-1}}{x^{a}}

is equal to a nonzero real number, is ______.

Q173·MathematicsIntegerJEE Advanced 2020

The value of the limit

\lim_{x \to \frac{\pi}{2}} \frac{4\sqrt{2}(\sin 3x + \sin x)}{\left( 2\sin 2x \sin\frac{3x}{2} + \cos\frac{5x}{2} \right) - \left( \sqrt{2} + \sqrt{2}\cos 2x + \cos\frac{3x}{2} \right)}

is ________

Q174·MathematicsNumericalJEE Main 2020

Let

f(x)=x.\left[\dfrac{x}{2}\right]

, for

-10<x<10

, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to ___

Q175·MathematicsSingle correctJEE Main 2020

\displaystyle\lim_{x\to 0}\dfrac{x\left(e^{\left(\sqrt{1+x^2+x^4}-1\right)/x}-1\right)}{\sqrt{1+x^2+x^4}-1}

(A)does not exist
(B)is equal to $1$
(C)is equal to $\sqrt{e}$
(D)is equal to $0$

Q176·MathematicsSingle correctJEE Main 2020

\lim_{x \to a} \frac{\left( a + 2x \right)^{\frac{1}{3}} - \left( 3x \right)^{\frac{1}{3}}}{\left( 3a + x \right)^{\frac{1}{3}} - \left( 4x \right)^{\frac{1}{3}}} \left( a \ne 0 \right)

is equal to:

(A) $\left( \frac{2}{3} \right)^{\frac{4}{3}}$
(B) $\left( \frac{2}{9} \right)^{\frac{4}{3}}$
(C) $\left( \frac{2}{9} \right) \left( \frac{2}{3} \right)^{\frac{1}{3}}$
(D) $\left( \frac{2}{3} \right) \left( \frac{2}{9} \right)^{\frac{1}{3}}$

Q177·MathematicsSingle correctJEE Main 2020

Let

[t]

denote the greatest integer

\leq t

. If for some

\lambda \in R - \{0, 1\}

,

\lim\limits_{x \to 0} \left| \frac{1 - x + |x|}{\lambda - x + [x]} \right| = L

, then L is equal to:

(A)0
(B)1
(C) $\frac{1}{2}$
(D)2

Q178·MathematicsNumericalJEE Main 2020

If

\lim\limits_{x \to 0} \left\{ \frac{1}{x^{8}} \left( 1 - \cos\frac{x^{2}}{2} - \cos\frac{x^{2}}{4} + \cos\frac{x^{2}}{2}\cos\frac{x^{2}}{4} \right) \right\} = 2^{-k}

, then the value of k is __________.

Q179·MathematicsSingle correctJEE Main 2020

Let

[t]

denote the greatest integer

\leq t

and

\lim\limits_{x \rightarrow 0} x\left[\dfrac{4}{x}\right] = A

. Then the function,

f(x) = \left[x^{2}\right]\sin(\pi x)

is discontinuous, when x is equal to:

(A) $\sqrt{A + 21}$
(B) $\sqrt{A + 5}$
(C) $\sqrt{A}$
(D) $\sqrt{A + 1}$

Q180·MathematicsSingle correctJEE Main 2020

\displaystyle\lim_{x\to 0}\dfrac{\int_{0}^{x} t\sin(10t)\,dt}{x}

is equal to:

(A)0
(B) $\dfrac{1}{10}$
(C) $-\dfrac{1}{5}$
(D) $-\dfrac{1}{10}$

Q181·MathematicsSingle correctJEE Main 2020

\displaystyle\lim_{x\to 0}\left(\dfrac{3x^{2}+2}{7x^{2}+2}\right)^{1/x^{2}}

is equal to:

(A) $\dfrac{1}{e}$
(B) $e$
(C) $e^{2}$
(D) $\dfrac{1}{e^{2}}$

Q182·MathematicsNumericalJEE Main 2020

If the function f defined on

\left(-\dfrac{1}{3},\dfrac{1}{3}\right)

by

f(x)=\begin{cases}\dfrac{1}{x}\log_{e}\left(\dfrac{1+3x}{1-2x}\right), & \text{when } x \ne 0 \\ k, & \text{when } x=0\end{cases}

is continuous, then k is equal to __________.

Q183·MathematicsNumericalJEE Main 2020

\lim\limits_{x \to 2} \frac{3^{x}+3^{3-x}-12}{3^{-x/2}-3^{1-x}}

is equal to _________

Q184·MathematicsSingle correctJEE Main 2019

Let

f(x) = 5 - |x-2|

and

g(x) = |x+1|

,

x \in R

. If

f(x)

attains maximum value at

\alpha

and

g(x)

attains minimum value at

\beta

, then

\displaystyle\lim_{x \to -\alpha\beta} \frac{(x-1)(x^{2}-5x+6)}{x^{2}-6x+8}

is equal to :

(A) $\dfrac{3}{2}$
(B) $\dfrac{-3}{2}$
(C) $\dfrac{1}{2}$
(D) $\dfrac{-1}{2}$

Q185·MathematicsSingle correctJEE Main 2019

\displaystyle\lim_{x\to 0} \dfrac{x + 2\sin x}{\sqrt{x^{2}+2\sin x+1} - \sqrt{\sin^{2}x - x + 1}}

is:

(A)2
(B)6
(C)3
(D)1

Q186·MathematicsSingle correctJEE Main 2019

If

\displaystyle\lim_{x\to 1}\dfrac{x^{4}-1}{x-1}=\lim_{x\to k}\dfrac{x^{3}-k^{3}}{x^{2}-k^{2}}

, then k is

(A) $\dfrac{3}{8}$
(B) $\dfrac{8}{3}$
(C) $\dfrac{4}{3}$
(D) $\dfrac{3}{2}$

Q187·MathematicsSingle correctJEE Main 2019

If the function

f\left(x\right)=\begin{cases} a\left|\pi-x\right|+1, & x\leq 5 \\ b\left|\pi-x\right|+3, & x>5 \end{cases}

is continuous at

x=5

, then the value of

a-b

is

(A) $\dfrac{2}{5-\pi}$
(B) $\dfrac{2}{\pi-5}$
(C) $\dfrac{2}{\pi+5}$
(D) $\dfrac{-2}{\pi+5}$

Q188·MathematicsSingle correctJEE Main 2019

If

f\left(x\right)=\left[x\right]-\left[\dfrac{x}{4}\right],x\in R

, where [x] denotes the greatest integer function, then:

(A)Both $\lim\limits_{x\to4-}f\left(x\right)$ and $\lim\limits_{x\to4+}f\left(x\right)$ exist but are not equal
(B) $\lim\limits_{x\to4-}f\left(x\right)$ exists but $\lim\limits_{x\to4+}f\left(x\right)$ does not exist
(C) $\lim\limits_{x\to4+}f\left(x\right)$ exists but $\lim\limits_{x\to4-}f\left(x\right)$ does not exist
(D)f is continuous at x = 4

Q189·MathematicsSingle correctJEE Main 2019

If the function f defined on

\left(\dfrac{\pi}{6}, \dfrac{\pi}{3}\right)

by

f(x)=\begin{cases}\dfrac{\sqrt{2}\cos x-1}{\cot x-1}, & x\neq \dfrac{\pi}{4}\\ k, & x=\dfrac{\pi}{4}\end{cases}

is continuous, then k is equal to:

(A)1
(B)2
(C) $\dfrac{1}{2}$
(D) $\dfrac{1}{\sqrt{2}}$

Q190·MathematicsSingle correctJEE Main 2019

If

f:R \rightarrow R

is a differentiable function and

f(2)=6

, then

\displaystyle\lim_{x\to 2}\int_{6}^{f(x)}\dfrac{2t\,dt}{(x-2)}

is:

(A) $0$
(B) $2f'(2)$
(C) $12f'(2)$
(D) $24f'(2)$

Q191·MathematicsSingle correctJEE Main 2019

Let

f:R\rightarrow R

be a differentiable function satisfying

f''(3)+f'(2)=0

. Then

\lim_{x\rightarrow\infty}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}

is equal to:

(A) $e^{2}$
(B)1
(C)e
(D) $e^{-1}$

Q192·MathematicsSingle correctJEE Main 2019

\displaystyle\lim_{x\to 0}\frac{\sin^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}

equals

(A) $\sqrt{2}$
(B) $4\sqrt{2}$
(C) $4$
(D) $2\sqrt{2}$

Q193·MathematicsSingle correctJEE Main 2019

\lim_{x\to 1^{-}} \dfrac{\sqrt{\pi} - \sqrt{2\sin^{-1} x}}{\sqrt{1-x}}

is equal to :

(A) $\dfrac{1}{\sqrt{2\pi}}$
(B) $\sqrt{\dfrac{2}{\pi}}$
(C) $\sqrt{\dfrac{\pi}{2}}$
(D) $\sqrt{\pi}$

Q194·MathematicsSingle correctJEE Main 2019

\lim_{x\to \pi/4}\frac{\cot^{3}x-\tan x}{\cos\left(x+\pi/4\right)}

is:

(A)4
(B) $4\sqrt{2}$
(C) $8\sqrt{2}$
(D)8

Q195·MathematicsSingle correctJEE Main 2019

\displaystyle\lim_{x\to 0}\frac{x\cot(4x)}{\sin^{2}x\cot^{2}(2x)}

is equal to:

(A)0
(B)2
(C)4
(D)1

Q196·MathematicsSingle correctJEE Main 2019

Let [x] denote the greatest integer less than or equal to x. Then:

\lim_{x\to 0}\frac{\tan\left(\pi\sin^{2}x\right)+\left(|x|-\sin\left(x[x]\right)\right)^{2}}{x^{2}}

(A)does not exist
(B)equals π
(C)equal π + 1
(D)equals 0

Q197·MathematicsSingle correctJEE Main 2019

\displaystyle\lim_{y \to 0} \dfrac{\sqrt{1+\sqrt{1+y^4}}-\sqrt2}{y^4} =

(A)exists and equals $\dfrac{1}{4\sqrt2}$
(B)exists and equals $\dfrac{1}{2\sqrt2(\sqrt2+1)}$
(C)exists and equals $\dfrac{1}{2\sqrt2}$
(D)does not exist

Q198·MathematicsSingle correctJEE Main 2019

Let

f:R \rightarrow R

be a function defined as

f(x)=\begin{cases} 5, & \text{if } x \le 1 \\ a+bx, & \text{if } 1 < x < 3 \\ b+5x, & \text{if } 3 \le x < 5 \\ 30, & \text{if } x \ge 5 \end{cases}

. Then

f

is:

(A)continuous if $a = 5$ and $b = 5$
(B)continuous if $a = 5$ and $b = 10$
(C)continuous if $a = 0$ and $b = 5$
(D)not continuous for any values of $a$ and $b$

Q199·MathematicsNumericalJEE Advanced 2018

For each positive integer n, let

y_{n} = \frac{1}{n}\left((n+1)(n+2)....(n+n)\right)^{1/n}

. For

x \in R

, let [x] be the greatest integer less than or equal to x. If

\lim_{n \to \infty} y_{n} = L

, then the value of [L] is ______ .

Q200·MathematicsMultiple correctJEE Advanced 2017

Let

[x]

be the greatest integer less than or equals to

x

. Then, at which of the following point(s) the function

f(x) = x \cos(\pi(x + [x]))

is discontinuous ?

(A) $x = -1$
(B) $x = 0$
(C) $x = 2$
(D) $x = 1$

Q201·MathematicsMultiple correctJEE Advanced 2017

Let

f(x) = \dfrac{1 - x\left(1 + |1-x|\right)}{|1-x|}\cos\left(\dfrac{1}{1-x}\right)

for

x \ne 1

. Then

(A) $\lim\limits_{x \to 1^{-}} f(x) = 0$
(B) $\lim\limits_{x \to 1^{-}} f(x)$ does not exist
(C) $\lim\limits_{x \to 1^{+}} f(x) = 0$
(D) $\lim\limits_{x \to 1^{+}} f(x)$ does not exist

Q202·MathematicsIntegerJEE Advanced 2016

Let

\alpha, \beta \in \mathbb{R}

be such that

\lim_{x \to 0} \frac{x^{2}\sin(\beta x)}{\alpha x - \sin x} = 1

. Then

6(\alpha + \beta)

equals

Q203·MathematicsIntegerJEE Advanced 2015

Let m and n be two positive integers greater than 1. If

\displaystyle\lim_{\alpha \to 0}\left(\dfrac{e^{\cos(\alpha^{n})} - e}{\alpha^{m}}\right) = -\left(\dfrac{e}{2}\right)

then the value of

\dfrac{m}{n}

is

Q204·MathematicsMultiple correctJEE Advanced 2014

For every pair of continuous functions

f, g: [0, 1] \to \mathbb{R}

such that

\max\{ f(x) : x \in [0, 1]\} = \max\{g(x) : x \in [0, 1]\}

, the correct statement(s) is(are)

(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0, 1]$
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0, 1]$
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0, 1]$
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0, 1]$

Q205·MathematicsIntegerJEE Advanced 2014

The largest value of the non-negative integer

a

for which

\lim_{x \to 1} \left\{\frac{-ax + \sin(x - 1) + a}{x + \sin(x - 1) - 1}\right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}

is __________

Q206·MathematicsMultiple correctJEE Advanced 2013

For

a \in \mathbb{R}

(the set of all real numbers),

a \neq -1

,

\lim_{n \to \infty} \frac{\left(1^{a} + 2^{a} + \ldots + n^{a}\right)}{(n+1)^{a-1}\left[(na+1) + (na+2) + \ldots + (na+n)\right]} = \frac{1}{60}

. Then

a =

(A) $5$
(B) $7$
(C) $\frac{-15}{2}$
(D) $\frac{-17}{2}$

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