Limits, Continuity and Differentiability PYQs - Last 10 Years
MHT CET / Mathematics / Calculus / 159 recent questions
MathematicsCalculus2017-2026
Practice 159 MHT CET 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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Last 10 Years Limits, Continuity and Differentiability Questions
Showing 50 of 159 filtered questions.
1Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 1}\left(\log _3 3 x\right)^{\log _x 8}=\ldots\)
MCQ+2 / -02025
2Limits Continuity And Differentiability
If $f(x)= \begin{cases}\frac{8^x-4^x-2^x+1}{x^2}, & \text { if } x>0 \\ e^x \sin x+x+\lambda \log 4, & \text { if } x \leqslant 0\end{cases}$
is continuous at $x=0$ then the value of $1000 \mathrm{e}^\lambda=$
is continuous at $x=0$ then the value of $1000 \mathrm{e}^\lambda=$
MCQ+2 / -02025
3Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to \infty } \frac{\mathrm{e}^{x^4}-1}{\mathrm{e}^{x^4}+1}=\)
MCQ+2 / -02025
4Limits Continuity And Differentiability
If $f(\theta)=\cos \theta_1 \cdot \cos \theta_2 \cdot \cos \theta_3$ .............. $\cos \theta_n$, then $\tan \theta_1+\tan \theta_2+\tan \theta_3+$. ............ $+\tan \theta_{\mathrm{n}}=$
MCQ+2 / -02025
5Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{n \to \infty }\left[\frac{1}{1-n^4}+\frac{8}{1-n^4}+\ldots \ldots \ldots \ldots .+\frac{n^3}{1-n^4}\right]=\)
MCQ+2 / -02025
6Limits Continuity And Differentiability
If $f(x)= \begin{cases}\frac{8^x-4^x-2^x+1^x}{x^2}, & \text { if } x>0 \\ \mathrm{e}^x \sin x+\mathrm{i} x+\lambda \log 4, & \text { if } x \leqslant 0, \mathrm{i} \in \mathbb{R}\end{cases}$ continuous at $x=0$, then the value of $500 \math...
MCQ+2 / -02025
7Limits Continuity And Differentiability
If $\mathrm{f}(x)=\left\{\begin{array}{ll}\operatorname{m} x+1, & x \leqslant \frac{\pi}{2} \\ \sin x+\mathrm{n}, & x>\frac{\pi}{2}\end{array}\right.$, is continuous at $x=\frac{\pi}{2},(\mathrm{~m}, \mathrm{n} \in \mathbb{Z})$ then
MCQ+2 / -02025
8Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \frac{\mathrm{e}^{\tan x}-\mathrm{e}^x}{\tan x-x}=\)
MCQ+2 / -02025
9Limits Continuity And Differentiability
$$ f(x)= \begin{cases}{\left[x^2\right]-\left[-x^2\right],} & x \neq 3 \\ k & , x=3\end{cases} $$
is continuous at $x=3$, then $\mathrm{k}=$ where $[\cdot]$ is greatest integer function
is continuous at $x=3$, then $\mathrm{k}=$ where $[\cdot]$ is greatest integer function
MCQ+2 / -02025
10Limits Continuity And Differentiability
$\mathop {\lim }\limits_{x \to 0} \frac{\left(7^x-1\right)^4}{\tan \left(\frac{x}{\mathrm{k}}\right) \cdot \log \left(1+\frac{x^2}{3}\right) \cdot \sin 4 x}=3(\log 7)^3$, then $\mathrm{k}=$
MCQ+2 / -02025
11Limits Continuity And Differentiability
$\lim _\limits{x \rightarrow 3} \frac{(84-x)^{\frac{1}{4}}-3}{x-3}$ is
MCQ+2 / -02025
12Limits Continuity And Differentiability
If $\mathrm{f}(x)$ is continuous at point $x=0$ where
$$ f(x)=\left\{\begin{array}{cc} \frac{3 \sin x+5 \tan x}{\mathrm{a}^x-1} & , x<0 \\ \frac{2}{\log 2} & , x=0 \\ \frac{8 x+2 x \cos x}{\mathrm{~b}^x-1} & , x>0 \end{array}\right. $$
th...
$$ f(x)=\left\{\begin{array}{cc} \frac{3 \sin x+5 \tan x}{\mathrm{a}^x-1} & , x<0 \\ \frac{2}{\log 2} & , x=0 \\ \frac{8 x+2 x \cos x}{\mathrm{~b}^x-1} & , x>0 \end{array}\right. $$
th...
MCQ+2 / -02025
13Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \frac{|x|}{|x|+x^2}=\)
MCQ+2 / -02025
14Limits Continuity And Differentiability
If $\mathrm{f}(x)=\frac{(27-2 x)^{\frac{1}{3}}-3}{9-3(243+5 x)^{\frac{1}{5}}}, x \neq 0$ is continuous at $x=0$, then the value of $\mathrm{f}(0)$ is
MCQ+2 / -02025
15Limits Continuity And Differentiability
Let k be a non-zero real number.
If $f(x)=\left\{\begin{array}{cl}\frac{\left(\mathrm{e}^x-1\right)^2}{\sin \left(\frac{x}{k}\right) \log \left(1+\frac{x}{4}\right)} & , x \neq 0 \\ 12 & , x=0\end{array}\right.$
is a continuous function, th...
If $f(x)=\left\{\begin{array}{cl}\frac{\left(\mathrm{e}^x-1\right)^2}{\sin \left(\frac{x}{k}\right) \log \left(1+\frac{x}{4}\right)} & , x \neq 0 \\ 12 & , x=0\end{array}\right.$
is a continuous function, th...
MCQ+2 / -02024
16Limits Continuity And Differentiability
If $\lim\limits_{x \rightarrow \infty}\left(\frac{x^2+x+1}{x+1}-a x-b\right)=4$ then
MCQ+2 / -02024
17Limits Continuity And Differentiability
The approximate value of $(3.978)^{3 / 2}$ is
MCQ+2 / -02024
18Limits Continuity And Differentiability
Let $\mathrm{f}(x)=x\left[\frac{x}{2}\right]$, for $-10< x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to
MCQ+2 / -02024
19Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow \frac{\pi}{2}} \frac{(1-\sin x)\left(8 x^3-\pi^3\right) \cos x}{(\pi-2 x)^4}\)
MCQ+2 / -02024
20Limits Continuity And Differentiability
$\lim _\limits{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x}$ has the value
MCQ+2 / -02024
21Limits Continuity And Differentiability
Let $a, b \in(a \neq 0)$. If the function $f$ is defined as
$$f(x)=\left\{\begin{array}{cc}
\frac{2 x^2}{\mathrm{a}} & , 0 \leq x<1 \\
\mathrm{a} & , 1 \leq x<\sqrt{2} \\
\frac{2 \mathrm{~b}^2-4 b}{x} & , \sqrt{2} \leq x<\infty
\end{array}\...
$$f(x)=\left\{\begin{array}{cc}
\frac{2 x^2}{\mathrm{a}} & , 0 \leq x<1 \\
\mathrm{a} & , 1 \leq x<\sqrt{2} \\
\frac{2 \mathrm{~b}^2-4 b}{x} & , \sqrt{2} \leq x<\infty
\end{array}\...
MCQ+2 / -02024
22Limits Continuity And Differentiability
If $f(x)=\left(\sin ^4 x+\cos ^4 x\right), 0< x<\frac{\pi}{2}$, then the function has minimum value at $x=$
MCQ+2 / -02024
23Limits Continuity And Differentiability
If $\mathrm{f}(x)=\frac{x+x^2+x^3+\ldots \ldots \ldots \ldots+x^{\mathrm{n}}-\mathrm{n}}{x-1}$, for $x \neq 1$ is continuous at $x=1$, then $\mathrm{f}(1)=$
MCQ+2 / -02024
24Limits Continuity And Differentiability
If $\lim _\limits{x \rightarrow 1} \frac{x^2-a x+b}{x-1}=7$, then $a+b$ is equal to
MCQ+2 / -02024
25Limits Continuity And Differentiability
Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation \((\sqrt[3]{1+a}-1) x^2+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0\) where $a>-1$ then $\lim _\limits{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _\limits{a \rightarrow 0^{+}} \beta(a)$ re...
MCQ+2 / -02024
26Limits Continuity And Differentiability
Let $\mathrm{f}(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{1}{2}\right], \quad \mathrm{f}(x)$ is continuous in $\left[0, \frac{\pi}{2}\right]$, then $\mathrm{f}\left(\frac{\pi}{4}\right)$ is
MCQ+2 / -02024
27Limits Continuity And Differentiability
For each $x \in \mathbb{R}$, Let $[x]$ represent greatest integer function, then $\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$ is equal to
MCQ+2 / -02024
28Limits Continuity And Differentiability
If the function $\mathrm{f}(x)=\left(\frac{5 x-8}{8-3 x}\right)^{\frac{3}{2 x-4}}$ if $x \neq 2$.
$=\mathrm{k}$
if $x=2$.
is continuous at $x=2$, then $\mathrm{k}=$
$=\mathrm{k}$
if $x=2$.
is continuous at $x=2$, then $\mathrm{k}=$
MCQ+2 / -02024
29Limits Continuity And Differentiability
If $\mathrm{f}(x)=\frac{1+\cos \pi x}{\pi(1-x)^2}$, for $x \neq 1$ is continuous at $x=1$, then $\mathrm{f}(1)$ is equal to
MCQ+2 / -02024
30Limits Continuity And Differentiability
Let K be the set of all real values of $x$, where the function $\mathrm{f}(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set K is
MCQ+2 / -02024
31Limits Continuity And Differentiability
The value of $\lim _\limits{x \rightarrow 0} \frac{x}{|x|+x^2}$ is
MCQ+2 / -02024
32Limits Continuity And Differentiability
$\lim _\limits{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ is equal to
MCQ+2 / -02024
33Limits Continuity And Differentiability
The value of k , for which the function
$$\mathrm{f}(x)= \begin{cases}\left(\frac{4}{5}\right)^{\frac{\ln 4 x}{\tan 5 x}}, & 0< x< \frac{\pi}{2} \\ \mathrm{k}+\frac{2}{5} & , x=\frac{\pi}{2}\end{cases}$$
is continuous at $x=\frac{\pi}{2}$, ...
$$\mathrm{f}(x)= \begin{cases}\left(\frac{4}{5}\right)^{\frac{\ln 4 x}{\tan 5 x}}, & 0< x< \frac{\pi}{2} \\ \mathrm{k}+\frac{2}{5} & , x=\frac{\pi}{2}\end{cases}$$
is continuous at $x=\frac{\pi}{2}$, ...
MCQ+2 / -02024
34Limits Continuity And Differentiability
Let $\mathrm{f}(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right]$. $f(x)$ is continuous in $\left[0, \frac{\pi}{2}\right]$, then $f\left(\frac{\pi}{4}\right)$ is
MCQ+2 / -02024
35Limits Continuity And Differentiability
The value of $\lim _\limits{x \rightarrow 0}\left((\sin x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right)$, where $x>0$ is
MCQ+2 / -02024
36Limits Continuity And Differentiability
If the function $f$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by
$$f(x)=\left\{\begin{array}{cc} \frac{\sqrt{2} \cos x-1}{\cot x-1}, & x \neq \frac{\pi}{4} \\ k \quad, & x=\frac{\pi}{4} \end{array}\right.$$
is continuous, then ...
$$f(x)=\left\{\begin{array}{cc} \frac{\sqrt{2} \cos x-1}{\cot x-1}, & x \neq \frac{\pi}{4} \\ k \quad, & x=\frac{\pi}{4} \end{array}\right.$$
is continuous, then ...
MCQ+2 / -02024
37Limits Continuity And Differentiability
\(\lim _\limits{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}=\)
MCQ+2 / -02024
38Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow \frac{\pi}{2}} \frac{\left(1-\tan \left(\frac{x}{2}\right)\right)(1-\sin x)}{\left(1+\tan \left(\frac{x}{2}\right)\right)(\pi-2 x)^3}\)
is
is
MCQ+2 / -02024
39Limits Continuity And Differentiability
Let $f:[-1,3] \rightarrow \mathbb{R}$ be defined as
$$\left\{\begin{array}{lc} |x|+[x], & -1 \leqslant x<1 \\ x+|x|, & 1 \leqslant x<2 \\ x+[x], & 2 \leqslant x \leqslant 3 \end{array}\right.$$
where $[t]$ denotes the greatest integer funct...
$$\left\{\begin{array}{lc} |x|+[x], & -1 \leqslant x<1 \\ x+|x|, & 1 \leqslant x<2 \\ x+[x], & 2 \leqslant x \leqslant 3 \end{array}\right.$$
where $[t]$ denotes the greatest integer funct...
MCQ+2 / -02024
40Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}=\)
MCQ+2 / -02024
41Limits Continuity And Differentiability
If the function $f(x)= \begin{cases}-2 \sin x & \text {, if } x \leq \frac{-\pi}{2} \\ A \sin x+B & , \text { if } \frac{-\pi}{2}< x<\frac{\pi}{2} \\ \cos x & , \text { if } x \geq \frac{\pi}{2}\end{cases}$ is continuous everywhere, then th...
MCQ+2 / -02024
42Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}\right)=\)
MCQ+2 / -02024
43Limits Continuity And Differentiability
The approximate value of $(3.978)^{\frac{3}{2}}$ is
MCQ+2 / -02024
44Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cc}\frac{a}{2}(x-|x|) & , \\ 0, & \text { for } x<0 \\ 0, & \text { for } x=0 \\ b x^2 \sin \left(\frac{1}{x}\right) & \text { for } x>0\end{array}\right.$
is continuous at $x=0$, then
is continuous at $x=0$, then
MCQ+2 / -02024
45Limits Continuity And Differentiability
Let f be twice differentiable function such that $\mathrm{f}^{\prime \prime}(x)=-\mathrm{f}(x), \mathrm{f}^{\prime}(x)=\mathrm{g}(x)$ and $\mathrm{h}(x)=(\mathrm{f}(x))^2+(\mathrm{g}(x))^2$. If $\mathrm{h}(5)=1$, then the value of $h(10)$ i...
MCQ+2 / -02024
46Limits Continuity And Differentiability
Let $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , x<0 \\ a & , x=0 \\ \frac{\sqrt{2}}{\sqrt{16+\sqrt{x-4}}} & , x>0\end{array}\right.$
If $\mathrm{f}(x)$ is continuous at $x=0$, then the value of $a$ is
If $\mathrm{f}(x)$ is continuous at $x=0$, then the value of $a$ is
MCQ+2 / -02024
47Limits Continuity And Differentiability
$\lim _\limits{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}$ is
MCQ+2 / -02024
48Limits Continuity And Differentiability
The values of $a$ and $b$, so that the function
$$f(x)= \begin{cases}x+\mathrm{a} \sqrt{2} \sin x & , 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b & , \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ \mathrm{a} \cos 2 x-\mathrm{b} \sin x & , \frac...
$$f(x)= \begin{cases}x+\mathrm{a} \sqrt{2} \sin x & , 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b & , \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ \mathrm{a} \cos 2 x-\mathrm{b} \sin x & , \frac...
MCQ+2 / -02024
49Limits Continuity And Differentiability
The approximate value of $x^3-2 x^2+3 x+2$ at $x=2.01$ is
MCQ+2 / -02024
50Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 0} \frac{9^x-4^x}{x\left(9^x+4^x\right)}=\)
MCQ+2 / -02024