Limits, Continuity and Differentiability PYQs - Last 5 Years
JEE Advanced / Mathematics / Calculus / 15 recent questions
MathematicsCalculus2022-2026
Practice 15 JEE Advanced Mathematics questions from Limits, Continuity and Differentiability. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.
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Mathematics / Calculus
2022-2026
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MCQ46.7%
INTEGER33.3%
MCQM20%
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15 in last 5 years15 in last 10 years
Last 5 Years Limits, Continuity and Differentiability Questions
Showing 15 of 15 filtered questions.
1Limits Continuity And Differentiability
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) \rightarrow(-\infty, \inft...
Consider the functions $f:(-3,3) \rightarrow(-\infty, \inft...
INTEGER+4 / -02026
2Limits Continuity And Differentiability
Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by\(f(x) = (|x| + |x-1|) \sin x + \left[ x \sin x \right],\)where $\left[ x \sin x \right]$ is the greatest integer less than or equal to $x \...
INTEGER+4 / -02026
3Limits Continuity And Differentiability
Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be an arbitrary function and let $g : \mathbb{R} \to \mathbb{R}$ be the function defined by\(g(x) = x f(x), \quad \text{for all } x \in \mathbb{R}.\)Th...
MCQM+4 / -12026
4Limits Continuity And Differentiability
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 T...
MCQ+3 / -12025
5Limits Continuity And Differentiability
Let $\mathbb{R}$ 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 entry in List-II and cho...
Match each entry in List-I to the correct entry in List-II and cho...
MCQ+4 / -12025
6Limits Continuity And Differentiability
Let α and β be the real numbers such that
$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $
Then the value of α + β is ___________.
$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $
Then the value of α + β is ___________.
INTEGER+4 / -02025
7Limits Continuity And Differentiability
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 ...
MCQ+3 / -12025
8Limits Continuity And Differentiability
Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that
$$ \lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\l...
$$ \lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\l...
MCQM+4 / -22024
9Limits Continuity And Differentiability
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 follow...
$$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$
Then which of the follow...
MCQ+3 / -12024
10Limits Continuity And Differentiability
Let $k \in \mathbb{R}$. If $\lim \limits_{x \rightarrow 0+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6$, then the value of $k$ is
MCQ+3 / -12024
11Limits Continuity And Differentiability
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by
$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x...
$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x...
MCQ+3 / -12024
12Limits Continuity And Differentiability
Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements i...
MCQM+4 / -22023
13Limits Continuity And Differentiability
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 \to...
\(f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}} .\)
Then, the value of $$\mathop {\lim }\limits_{n \to...
MCQ+3 / -12022
14Limits Continuity And Differentiability
If
\(\beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},\)
then the value of $6 \beta$ is ___________.
\(\beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},\)
then the value of $6 \beta$ is ___________.
INTEGER+3 / -12022
15Limits Continuity And Differentiability
Let \(\alpha\) be a positive real number. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g:(\alpha, \infty) \rightarrow \mathbb{R}\) be the functions defined by
$$
f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=...
$$
f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=...
INTEGER+3 / -02022