Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define $$g(x) = \frac{3x e^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}$$ for all real numbers $x$. Then which one of the following statements is TRUE?

For $\alpha = 3$, $\displaystyle \lim_{x \ o x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 0$

For $\alpha = 2$, $\displaystyle \lim_{x \ o x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 0$

For $\alpha = 2$, $\displaystyle \lim_{x \ o x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = 1$

For $\alpha = 3$, $\displaystyle \lim_{x \ o x_0} \left| \frac{g(x) + e^{x_0}}{x - x_0} \right| = \frac{2}{3}$

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by $f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$ Then which one of the following statements is TRUE?

For any positive real number $\delta$, the function $f$ is NOT an increasing function on the interval $(-\delta, 0)$

The function $f$ is NOT differentiable at $x = 0$

There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval $(0, \delta)$

$x = 0$ is a point of local minima of $f$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$ Then which of the following statements is TRUE?

$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.

$f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$.

$f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$.

The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite.

For positive integer $n$, define $$ f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}} . $$ Then, the value of $$\mathop {\lim }\limits_{n \ o \infty } f\left( n \right)$$ is equal to :

$4-\frac{3}{4} \log _{e}\left(\frac{7}{3}\right)$

$4-\frac{4}{3} \log _{e}\left(\frac{7}{3}\right)$

Limits Continuity And Differentiability — JEE Advanced Previous Year Questions

37 solved PYQs · 2016–2026 · Mathematics

Solved PYQs: 37
Years covered: 2016–2026
Avg / year: ~3
Subject: Mathematics

How many JEE Advanced questions come from Limits Continuity And Differentiability?

Limits Continuity And Differentiability has contributed 37 questions to Joint Entrance Examination (Advanced) papers between 2016 and 2026 — an average of about 3 questions a year. That count is measured from EByAi’s bank of 18,502 JEE Advanced questions, not estimated from a syllabus. Every one of the 37 questions carries a written, step-by-step solution, and each is tagged to Limits Continuity And Differentiability within Mathematics, so you can drill this chapter across every year at once instead of hunting through paper PDFs. Practising a chapter’s full PYQ history is the fastest way to learn the patterns examiners reuse.

Sample Limits Continuity And Differentiability questions from real JEE Advanced papers

These are 5 real questions from the bank — exactly as asked, with the answer one tap away. All 37 Limits Continuity And Differentiability questions (with full written solutions) are available in practice.

Question 1 · JEE Advanced 2025
Let $x_{0}$ be the real number such that $e^{x_{0}} + x_{0} = 0$ . For a given real number $α$ , define $g (x) = \frac{3 x e ^{x} + 3 x - α e ^{x} - α x}{3 ( e ^{x} + 1 )}$ for all real numbers $x$ . Then which one of the following statements is TRUE?
1. (A) $For α = 2, x o x_{0} lim \frac{g ( x ) + e ^{x_{0}}}{x - x _{0}} = 0$
2. (B) $For α = 2, x o x_{0} lim \frac{g ( x ) + e ^{x_{0}}}{x - x _{0}} = 1$
3. (C) $For α = 3, x o x_{0} lim \frac{g ( x ) + e ^{x_{0}}}{x - x _{0}} = 0$
4. (D) $For α = 3, x o x_{0} lim \frac{g ( x ) + e ^{x_{0}}}{x - x _{0}} = \frac{2}{3}$
Show answerAnswer
$Answer: (C) For α = 3, x o x_{0} lim \frac{g ( x ) + e ^{x_{0}}}{x - x _{0}} = 0$
Question 2 · JEE Advanced 2025
Let $R$ denote the set of all real numbers. Define the function $f : R \to R$ by $f (x) = {2 - 2 x^{2} - x^{2} sin \frac{1}{x} 2 if x \neq = 0, if x = 0.$ Then which one of the following statements is TRUE?
1. (A) $The function f is NOT differentiable at x = 0$
2. (B) $There is a positive real number δ, such that f is a decreasing function on the interval (0, δ)$
3. (C) $For any positive real number δ, the function f is NOT an increasing function on the interval (- δ, 0)$
4. (D) $x = 0 is a point of local minima of f$
Show answerAnswer
$Answer: (C) For any positive real number δ, the function f is NOT an increasing function on the interval (- δ, 0)$
Question 3 · JEE Advanced 2024
Let $k \in R$ . If $x \to 0 + lim (sin (sin k x) + cos x + x)^{\frac{2}{x}} = e^{6}$ , then the value of $k$ is
1. (A) $1$
2. (B) $2$
3. (C) $3$
4. (D) $4$
Show answerAnswer
$Answer: (B) 2$
Question 4 · JEE Advanced 2024
Let $f : R \to R$ be a function defined by $f (x) = {x^{2} sin (\frac{π}{x ^{2}}), 0, if x \neq = 0, if x = 0.$ Then which of the following statements is TRUE?
1. (A) $f (x) = 0 has infinitely many solutions in the interval [\frac{1}{1 0 ^{10}}, \infty) .$
2. (B) $f (x) = 0 has no solutions in the interval [\frac{1}{π}, \infty) .$
3. (C) $The set of solutions of f (x) = 0 in the interval (0, \frac{1}{1 0 ^{10}}) is finite.$
4. (D) $f (x) = 0 has more than 25 solutions in the interval (\frac{1}{π ^{2}}, \frac{1}{π}) .$
Show answerAnswer
$Answer: (D) f (x) = 0 has more than 25 solutions in the interval (\frac{1}{π ^{2}}, \frac{1}{π}) .$
Question 5 · JEE Advanced 2022
For positive integer $n$ , define $f (n) = n + \frac{16 + 5 n - 3 n ^{2}}{4 n + 3 n ^{2}} + \frac{32 + n - 3 n ^{2}}{8 n + 3 n ^{2}} + \frac{48 - 3 n - 3 n ^{2}}{12 n + 3 n ^{2}} + \dots + \frac{25 n - 7 n ^{2}}{7 n ^{2}} .$ Then, the value of $n o \infty lim f (n)$ is equal to :
1. (A) $3 + \frac{4}{3} lo g_{e} 7$
2. (B) $4 - \frac{3}{4} lo g_{e} (\frac{7}{3})$
3. (C) $4 - \frac{4}{3} lo g_{e} (\frac{7}{3})$
4. (D) $3 + \frac{3}{4} lo g_{e} 7$
Show answerAnswer
$Answer: (B) 4 - \frac{3}{4} lo g_{e} (\frac{7}{3})$

Limits Continuity And Differentiability questions by year

Year	Questions	Share of chapter questions
2026	3
2025	4
2024	4
2023	1
2022	3
2021	1
2020	5
2019	3
2018	5
2017	5
2016	3

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FAQs

How many JEE Advanced questions have been asked from Limits Continuity And Differentiability?

EByAi's bank holds 37 real JEE Advanced questions from Limits Continuity And Differentiability, asked between 2016 and 2026 — about 3 per year on average. Every count is measured from actual papers, not estimated.

Is Limits Continuity And Differentiability important for JEE Advanced?

37 questions over 11 papers make Limits Continuity And Differentiability a steady-weightage chapter in JEE Advanced Mathematics. The year-wise table above shows exactly how often it appears.

Can I solve Limits Continuity And Differentiability PYQs online for free?

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