Limits, Continuity & Differentiability

25 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q10·MCQ·Moderate
Open
Limits of indeterminate forms
Let $f : \mathbb{R} \to (0, \infty)$ be a twice differentiable function such that $f(3) = 18$ , $f'(3) = 0$ and $f''(3) = 4$ . Then $\lim_{x\to1}\left(\log_e\left(\dfrac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to:
1. (1)1
2. (2)9
3. (3)2
4. (4)18
MathsJEE Main 21 Jan 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
Derivatives of a polynomial
Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$ , $x \in \mathbb{R}$ . Then the value of $f'(5)$ is:
1. (1) $\dfrac{62}{5}$
2. (2) $\dfrac{657}{5}$
3. (3) $\dfrac{2}{5}$
4. (4) $\dfrac{117}{5}$
MathsJEE Main 21 Jan 2026, Shift 2·Q21·Numerical·Tough
Open
Sandwich theorem with greatest integer
Let $[\cdot]$ denote the greatest integer function and $f(x) = \displaystyle\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^{n}\left[\frac{k^2}{3^x}\right]$ . Then $12\displaystyle\sum_{j=1}^{\infty} f(j)$ is equal to .
MathsJEE Main 22 Jan 2026, Shift 2·Q5·MCQ·Moderate
Open
Limits using series expansion
If $\displaystyle\lim_{x\to0}\frac{e^{(a-1)x} + 2\cos bx + (c-2)e^{-x}}{x\cos x - \log_e(1+x)} = 2$ , then $a^2 + b^2 + c^2$ is equal to:
1. (1)5
2. (2)3
3. (3)7
4. (4)9
MathsJEE Main 22 Jan 2026, Shift 2·Q15·MCQ·Moderate
Open
Discontinuity of greatest integer function
Let $[\,\bullet\,]$ denote the greatest integer function, and let $f(x) = \min\left\{\sqrt2 x, x^2\right\}$ . Let $S = \{x \in (-2, 2) : \text{the function } g(x) = \lvert x\rvert\left[x^2\right] \text{ is discontinuous at } x\}$ . Then $\displaystyle\sum_{x\in S} f(x)$ equals:
1. (1) $2 - \sqrt2$
2. (2) $2\sqrt6 - 3\sqrt2$
3. (3) $1 - \sqrt2$
4. (4) $\sqrt6 - 2\sqrt2$
MathsJEE Main 23 Jan 2026, Shift 1·Q9·MCQ·Moderate
Open
Continuity of piecewise function
Let $f(x) = \begin{cases} \dfrac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\dfrac32, \dfrac12\\[2ex] b, & x = -\dfrac32, \dfrac12\end{cases}$ be continuous at $x = -\dfrac32$ . If $f\circ f(x) = \dfrac75$ , then $x$ is equal to:
1. (1)2
2. (2)1
3. (3)0
4. (4)1.4
MathsJEE Main 23 Jan 2026, Shift 2·Q1·MCQ·Easy–moderate
Open
Continuity at a point
If $f(x) = \begin{cases} \dfrac{a\lvert x\rvert + x^2 - 2(\sin\lvert x\rvert)(\cos\lvert x\rvert)}{x}, & x \neq 0\\[2mm] b, & x = 0\end{cases}$ is continuous at $x = 0$ , then $a + b$ is equal to:
1. (1)1
2. (2)2
3. (3)0
4. (4)4
MathsJEE Main 24 Jan 2026, Shift 1·Q1·MCQ·Moderate
Open
Limits using series expansion
If the function $f(x) = \dfrac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x}$ is Continuous at $x = 0$ , then the value of $f(0)$ is equal to
1. (1)2
2. (2) $\dfrac23$
3. (3) $\dfrac12$
4. (4) $\dfrac32$
MathsJEE Main 24 Jan 2026, Shift 1·Q16·MCQ·Easy–moderate
Open
Differentiability of piecewise function
Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x) = \begin{cases} 2\alpha(x^2 - 2) + 2\beta x, & x < 1\\ (\alpha + 3)x + (\alpha - \beta), & x \ge 1\end{cases}$ be differentiable at all $x \in \mathbb{R}$ . Then $34(\alpha + \beta)$ is equal to
1. (1)84
2. (2)48
3. (3)36
4. (4)24
MathsJEE Main 24 Jan 2026, Shift 2·Q20·MCQ·Moderate
Open
Continuity of piecewise functions
Let $[t]$ denote the greatest integer less than or equal to $t$ . If the function $f(x) = \begin{cases} b^2\sin\left(\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x + \sin x)\cos x\right]\right), & x < 0\\[10pt] \dfrac{\sin x - \frac12\sin 2x}{x^3}, & x > 0\\[10pt] a, & x = 0\end{cases}$ is continuous at $x = 0$ , then $a^2 + b^2$ is equal to
1. (1) $\dfrac58$
2. (2) $\dfrac{9}{16}$
3. (3) $\dfrac34$
4. (4) $\dfrac12$
MathsJEE Main 28 Jan 2026, Shift 1·Q14·MCQ·Moderate
Open
Limits using standard expansions
The value of $\lim_{x\to0}\frac{\log_e\left(\sec(ex)\cdot\sec\left(e^2x\right)\cdot\ldots\cdot\sec\left(e^{10}x\right)\right)}{e^2 - e^{2\cos x}}$ is equal to
1. (1) $\dfrac{(e^{10} - 1)}{2e^2(e^2 - 1)}$
2. (2) $\dfrac{(e^{20} - 1)}{2e^2(e^2 - 1)}$
3. (3) $\dfrac{(e^{20} - 1)}{2(e^2 - 1)}$
4. (4) $\dfrac{(e^{10} - 1)}{2(e^2 - 1)}$
MathsJEE Main 28 Jan 2026, Shift 2·Q5·MCQ·Tough
Open
Continuity of a limit-defined function
Let $f(x) = \lim_{n\to\infty}\left(\dfrac{\cos\pi x - x^{2n}\sin(x-1)}{1 + x^{2n}(x-1)}\right),\ x \in \mathbb{R}.$ Consider the following two statements :
(I) $f(x)$ is discontinuous at $x = 1$ .
(II) $f(x)$ is continuous at $x = -1$ .
Then,
1. (1)Neither (I) nor (II) is True
2. (2)Both (I) and (II) are True
3. (3)Only (II) is True
4. (4)Only (I) is True
We corrected a printing error in the original paper
Source printed “discontinous” and “continous”; source printed the limit as $\theta\to0$ with $x^{(2/\theta)}$ in place of $x^{2n}$ (undefined for $x<0$ , and the one-sided limits in $\theta$ differ for $0<\lvert x\rvert<1$ ); corrected to the intended $n\to\infty$ with $x^{2n}$ , the reading the key uses.
MathsJEE Main 2 Apr 2026, Shift 1·Q16·MCQ·Moderate
Open
Limits of indeterminate forms
If $\displaystyle\lim_{x\to2}\frac{\sin\left(x^3 - 5x^2 + ax + b\right)}{\left(\sqrt{x-1} - 1\right)\log_e(x-1)} = m$ , then $a + b + m$ is equal to:
1. (1)5
2. (2)6
3. (3)8
4. (4)10
MathsJEE Main 2 Apr 2026, Shift 2·Q25·Numerical·Easy–moderate
Open
Discontinuity of greatest integer function
The number of points in the interval $[2, 4]$ , at which the function $f(x) = \left[x^2 - x - \dfrac12\right]$ , where $[\,\cdot\,]$ denotes the greatest integer function, is discontinuous, is .
MathsJEE Main 4 Apr 2026, Shift 1·Q16·MCQ·Moderate
Open
Derivative of inverse trigonometric functions
If $y = \tan^{-1}\left(\dfrac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\dfrac{x}{1 + \sqrt{1 - x^2}}\right)$ , then $\dfrac{dy}{dx}$ at $x = \dfrac{\sqrt3}{2}$ is equal to:
1. (1)3
2. (2) $-1$
3. (3)1
4. (4)2
MathsJEE Main 4 Apr 2026, Shift 1·Q17·MCQ·Moderate
Open
Polynomial satisfying a derivative identity
Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x)\,f''(x)$ , for all $x \in \mathbb{R}$ . If $f(0) = 0$ , then $36\left(f'(2) + f''(2) + \displaystyle\int_0^2 f(x)\,dx\right)$ is equal to:
1. (1)42
2. (2)46
3. (3)56
4. (4)66
MathsJEE Main 4 Apr 2026, Shift 1·Q25·Numerical·Moderate
Open
Points of non-differentiability
The number of points, at which the function $f(x) = \max\{6x, 2 + 3x^2\} + \lvert x - 1\rvert\cos\left\lvert x^2 - \dfrac14\right\rvert$ , $x \in (-\pi, \pi)$ , is not differentiable, is .
We corrected a printing error in the original paper
Source printed “fucntion”.
MathsJEE Main 4 Apr 2026, Shift 2·Q22·Numerical·Moderate
Open
Continuity and differentiability of modulus functions
Let $f(x) = \begin{cases} e^{x-1}, & x < 0\\ x^2 - 5x + 6, & x \ge 0\end{cases}$ and $g(x) = f(\lvert x\rvert) + \lvert f(x)\rvert$ . If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to .
MathsJEE Main 5 Apr 2026, Shift 1·Q17·MCQ·Moderate
Open
Trigonometric limits
The product of all possible values of $\alpha$ , for which $\lim_{x\to0}\left(\frac{1 - \cos(\alpha x)\cos\big((\alpha+1)x\big)\cos\big((\alpha+2)x\big)}{\sin^2\big((\alpha+1)x\big)}\right) = 2,$ is:
1. (1) $-2$
2. (2)1
3. (3) $-1$
4. (4) $\dfrac54$
MathsJEE Main 5 Apr 2026, Shift 1·Q19·MCQ·Moderate
Open
Functional equation and differentiability
Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f\left(\frac{x + y}{3}\right) = \frac{f(x) + f(y)}{3}\quad\text{for all } x, y \in \mathbb{R}$ and $f'(0) = 3$ . Then the minimum value of the function $g(x) = 3 + e^x f(x)$ , is
1. (1) $3\left(\dfrac{e + 1}{e}\right)$
2. (2) $3\left(\dfrac{e - 1}{e}\right)$
3. (3) $\dfrac{3 - e}{e}$
4. (4) $3e$
MathsJEE Main 5 Apr 2026, Shift 2·Q17·MCQ·Moderate
Open
Standard trigonometric limits
Let $f(x) = \displaystyle\lim_{y\to0}\frac{(1 - \cos(xy))\tan(xy)}{y^3}$ . Then the number of solutions of the equation $f(x) = \sin x$ , $x \in \mathbb{R}$ is:
1. (1)0
2. (2)2
3. (3)3
4. (4)1
MathsJEE Main 6 Apr 2026, Shift 1·Q17·MCQ·Easy–moderate
Open
Limits by series expansion
The value of $\displaystyle\lim_{x\to0}\left(\frac{x^2\sin^2x}{x^2 - \sin^2x}\right)$ is:
1. (1)2
2. (2)3
3. (3)4
4. (4)6
MathsJEE Main 6 Apr 2026, Shift 2·Q16·MCQ·Tough
Open
Evaluation of limits
Let $\lim_{x\to2}\dfrac{\left(\tan(x-2)\right)\left(rx^2 + (p-2)x - 2p\right)}{(x-2)^2} = 5$ for some $r, p \in \mathbb{R}$ . If the set of all possible values of $q$ , such that the roots of the equation $rx^2 - px + q = 0$ lie in $(0, 2)$ , be the interval $(\alpha, \beta]$ , then $4(\alpha + \beta)$ equals.
1. (1)11
2. (2)13
3. (3)17
4. (4)21
MathsJEE Main 6 Apr 2026, Shift 2·Q25·Numerical·Moderate
Open
Continuity of composite function
Let $f(x) = \begin{cases} x^3 + 8 & x < 0\\ x^2 - 4 & x \ge 0\end{cases} \quad\text{and}\quad g(x) = \begin{cases} (x - 8)^{1/3}; & x < 0\\ (x + 4)^{1/2}; & x \ge 0\end{cases}$ Then the number of points, where the function $g\mathrm{o}f$ is discontinuous, is .
MathsJEE Main 8 Apr 2026, Shift 2·Q13·MCQ·Moderate
Open
Differentiability of modulus functions
For the function $f(x) = e^{\sin\lvert x\rvert} - \lvert x\rvert$ , $x \in \mathbb{R}$ , consider the following statements:
Statement I: $f$ is differentiable for all $x \in \mathbb{R}$ .
Statement II: $f$ is increasing in $\left(-\pi, -\dfrac{\pi}{2}\right)$ .
In the light of the above statements, choose the correct answer from the options given below:
1. (1)Both Statement I and Statement II are true
2. (2)Both Statement I and Statement II are false
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
We corrected a printing error in the original paper
Answer corrected; the source key is (4) — for $f(x) = e^{\sin\lvert x\rvert} - \lvert x\rvert$ both one-sided derivatives at $x = 0$ are $0$ and $f$ is smooth elsewhere, so Statement I is true, and $f'(x) = 1 - \cos x\, e^{-\sin x} > 0$ on $(-\pi, -\pi/2)$ , so Statement II is true; the source's solution works with $e^{\lvert\sin x\rvert}$ instead.