Limits, Continuity and Differentiability PYQs - Last 10 Years
AP EAPCET / Mathematics / Calculus / 103 recent questions
MathematicsCalculus2016-2025
Practice 103 AP EAPCET 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 103 filtered questions.
1Limits Continuity And Differentiability
If $f(x)=\frac{5 x \cdot \operatorname{cosec}(\sqrt{x})-1}{(x-2) \operatorname{cosec}(\sqrt{x})}$, then $\lim \limits_{x \rightarrow \infty} f\left(x^2\right)=$
MCQ+1 / -02024
2Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow 2} \frac{\sqrt{1+4 x}-\sqrt{3+3 x}}{x^3-8}=$
MCQ+1 / -02024
3Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \left( {{{\sin (\pi {{\cos }^2}x} \over {{x^2}}}} \right) =\)
MCQ+1 / -02024
4Limits Continuity And Differentiability
If the function $f(x)=\frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$, then $f(0)=$
MCQ+1 / -02024
5Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 1} \left( {{{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}} \right) =\)
MCQ+1 / -02024
6Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow 1} \frac{\sqrt{x}-1}{\left(\cos ^{-1} x\right)^2}=$
MCQ+1 / -02024
7Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2+x^5+x^6}}{x^4}=$
MCQ+1 / -02024
8Limits Continuity And Differentiability
If a function $f(x)=\left\{\begin{array}{cl}\frac{\tan (\alpha+1) x+\tan 2 x}{x} & \text { if } x>0 \\ \beta & \text { at } x=0 \text { is } \\ \frac{\sin 3 x-\tan 3 x}{x^3} & \text { if } x<0\end{array}\right.$
continuous at $x=0$, then $...
continuous at $x=0$, then $...
MCQ+1 / -02024
9Limits Continuity And Differentiability
In the interval $[0,3]$ The function $f(x)=|x-1|+|x-2|$ is
MCQ+1 / -02024
10Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x}=$
MCQ+1 / -02024
11Limits Continuity And Differentiability
If $\lim \limits_{x \rightarrow 0} \frac{e^x-a-\log (1+x)}{\sin x}=0$, then $a=$
MCQ+1 / -02024
12Limits Continuity And Differentiability
The values of $a$ and $b$ for which the function$ f(x)=\left\{\begin{array}{cl}1+|\sin x|^{\frac{a}{\sin x \mid}} & \frac{-\pi}{6} < x < 0 \\ b, & x=0 \quad \text { is continuous at } x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x},} & 0 < x < \frac{\p...
MCQ+1 / -02024
13Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 1 \\ a x^2+b x, & x>1\end{array}\right.$
is differentiable, $\forall x \in R$, then $f^{\prime}(2)=$
is differentiable, $\forall x \in R$, then $f^{\prime}(2)=$
MCQ+1 / -02024
14Limits Continuity And Differentiability
Let $f(x)=\min \left\{x, x^2\right\}$ for every real number of $x$, then
MCQ+1 / -02024
15Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cl}x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$
which of the following is true?
which of the following is true?
MCQ+1 / -02024
16Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{ll}3 a x-2 b, & x>1 \\ a x+b+1, & x<1\end{array}\right.$ and
$\lim \limits_{x \rightarrow 1} f(x)$ exists, then the relation between $a$ and $b$ is
$\lim \limits_{x \rightarrow 1} f(x)$ exists, then the relation between $a$ and $b$ is
MCQ+1 / -02024
17Limits Continuity And Differentiability
\(\lim \limits_{x \rightarrow 3} \frac{x^3-27}{x^2-9}=\)
MCQ+1 / -02024
18Limits Continuity And Differentiability
The function $f(x)=\left\{\begin{array}{ll}\frac{2}{5-x}, & x<3 \\ 5-x, & x \geq 3\end{array}\right.$ is
MCQ+1 / -02024
19Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cl}\frac{2 x e^{1 / 2 x}-3 x e^{-1 / 2 x}}{e^{1 / 2 x}+4 e^{-1 / 2 x}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$ is a real valued function, then
MCQ+1 / -02024
20Limits Continuity And Differentiability
$f(x)=\left\{\begin{array}{cl}\frac{\left(2 x^2-a x+1\right)-\left(a x^2+3 b x+2\right)}{x+1}, & \text { if } x \neq-1 \\ k_k, & \text { if } x=-1\end{array}\right.$
is a real valued function. If $a, b, k \in R$ and $f$ is continuous on $R$...
is a real valued function. If $a, b, k \in R$ and $f$ is continuous on $R$...
MCQ+1 / -02024
21Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow-1}\left(\frac{3 x^2-2 x+3}{3 x^2+x-2}\right)^{3 x-2}=$
MCQ+1 / -02024
22Limits Continuity And Differentiability
$\lim \limits_{x \rightarrow 0} \frac{1-\cos x \cdot \cos 2 x}{\sin ^2 x}=$
MCQ+1 / -02024
23Limits Continuity And Differentiability
Let $f(x)$ be a real valued function. If $f^{\prime}(x)$ is a constant for all $x \in R, f(0)=2$ and $f^{\prime}(0)=1$, then
MCQ+1 / -02023
24Limits Continuity And Differentiability
If $f: R \rightarrow R$ defined by
$$ f(x)= \begin{cases}\frac{\sin x-\sin \frac{X}{2}}{x}, & x<0 \\ \frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{cases} $$
is continuous on $R$, then $f(0)=$
$$ f(x)= \begin{cases}\frac{\sin x-\sin \frac{X}{2}}{x}, & x<0 \\ \frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{cases} $$
is continuous on $R$, then $f(0)=$
MCQ+1 / -02023
25Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cl}1+6 x-3 x^2, & x \leq 1 \\ x+\log _2\left(b^2+7\right), & x>1\end{array}\right.$ is continuous at all real $x$, then $b=$
MCQ+1 / -02023
26Limits Continuity And Differentiability
The function $f(x)=\sqrt{\frac{3 x^2-5 x-2}{2 x^2-7 x+5}}$ has discontinuous points at $x=$
MCQ+1 / -02023
27Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cc}\frac{x-[x]}{x-2}, & x > 2 \\ b, & x=2 \\ \frac{\left|x^2-x-2\right|}{a\left(2+x-x^2\right)}, & -1 < x \leq 2 \\ 2 a-b, & x \leq-1\end{array}\right.$
is continuous on $R$, then $\lim _{x \rightarrow 0} \frac...
is continuous on $R$, then $\lim _{x \rightarrow 0} \frac...
MCQ+1 / -02023
28Limits Continuity And Differentiability
Let $[x]$ represents the greatest integer not more than $x$. The discontinuous points of the function $f(x)=\frac{5+[x]}{\sqrt{11+[x]-6 \sqrt{2+[x]}}}$ lies in the interval
MCQ+1 / -02023
29Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow-\infty} \frac{3|x|-x}{|x|-2 x}-\lim _\limits{x \rightarrow 0} \frac{\log \left(1+x^3\right)}{\sin ^3 x}=\)
MCQ+1 / -02022
30Limits Continuity And Differentiability
If \(a, b\) and \(c\) are three distinct real numbers and \(\lim _\limits{x \rightarrow \infty} \frac{(b-c) x^2+(c-a) x+(a-b)}{(a-b) x^2+(b-c) x+(c-a)}=\frac{1}{2}\), then \(a+2 c=\)
MCQ+1 / -02022
31Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow-\infty} \log _e(\cosh x)+x=\)
MCQ+1 / -02022
32Limits Continuity And Differentiability
If \(f(x)=\operatorname{Max}\{3-x, 3+x, 6\}\) is not differentiable at \(x=a\), and \(x=b\), then \(|a|+|b|=\)
MCQ+1 / -02022
33Limits Continuity And Differentiability
If \(\lim _\limits{n \rightarrow \infty} x^n \log _e x=0\), then \(\log _x 12=\)
MCQ+1 / -02022
34Limits Continuity And Differentiability
Let \(f: R^{+} \longrightarrow R^{+}\) be a function satisfying \(f(x)-x=\lambda\) (constant), \(\forall x \in R^{+}\) and \(f(x f(y))=f(x y)+x, \forall x, y, \in R^{+}\). Then, $$\lim _\limits{x \rightarrow 0} \frac{(f(x))^{1 / 3}-1}{(f(x)...
MCQ+1 / -02022
35Limits Continuity And Differentiability
\(\lim _\limits{n \rightarrow \infty}\left(\frac{1}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)=\)
MCQ+1 / -02022
36Limits Continuity And Differentiability
$$\begin{aligned}
& \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\
& \lim _{x \rightarrow 0} x^4 \sin \left(\frac{1}{3 \sqrt{x}}\right)=l \text {. Then, } k+l=
\end{aligned}$$
MCQ+1 / -02022
37Limits Continuity And Differentiability
If \(l, m(l< m)\) are roots of \(a x^2+b x+c=0\), then
\(\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=\)
\(\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=\)
MCQ+1 / -02022
38Limits Continuity And Differentiability
Let $$f(x)=\left\{\begin{array}{cl}\frac{1}{|x|}, & \text { for }|x|>1 \\ a x^2+b, & \text { for }|x| \leq 1\end{array}\right.$$. If \(\lim _\limits{x \rightarrow 1^{+}} f(x)\) and \(\lim _\limits{x \rightarrow 1^{-}} f(x)\) exist, then the...
MCQ+1 / -02022
39Limits Continuity And Differentiability
If $$f(x)=\left\{\begin{array}{cc}\frac{x^2 \log (\cos x)}{\log (1+x)} & , \quad x \neq 0 \\ 0 & , x=0\end{array}\right.$$, then at \(x=0, f(x)\) is
MCQ+1 / -02022
40Limits Continuity And Differentiability
\(\frac{d}{d x}\left(\lim _{x \rightarrow 2} \frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)=\)
MCQ+1 / -02022
41Limits Continuity And Differentiability
If \([\cdot]\) denotes greatest integer function, then \(\lim _\limits{x \rightarrow \frac{-3}{5}} \frac{1}{\dot{x}}\left[\frac{-1}{x}\right]=\)
MCQ+1 / -02022
42Limits Continuity And Differentiability
If \(f(x)\), defined below, is continuous at \(x=4\), then
$$f(x) = \left\{ {\matrix{
{{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr
{a + b} & , & {x = 4} \cr
{{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr
} } \right....
$$f(x) = \left\{ {\matrix{
{{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr
{a + b} & , & {x = 4} \cr
{{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr
} } \right....
MCQ+1 / -02021
43Limits Continuity And Differentiability
If the function \(f(x)\), defined below, is continuous on the interval \([0,8]\), then $$f(x)=\left\{\begin{array}{cc}x^2+a x+b & , \quad 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b & , 4 < x \leq 8\end{array}\right.$$
MCQ+1 / -02021
44Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{n \to \infty } {{n{{(2n + 1)}^2}} \over {(n + 2)({n^2} + 3n - 1)}}\) is equal to
MCQ+1 / -02021
45Limits Continuity And Differentiability
If \(f(x)=\frac{\log _e\left(1+x^2(\tan x)\right)}{\sin x^3}, x \neq 0\) is to be continuous at \(x=0\), then \(f(0)\) must be equal to
MCQ+1 / -02021
46Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 1} \frac{(1-x)\left(1-x^2\right) \ldots\left(1-x^{2 n}\right)}{\left\{(1-x)\left(1-x^2\right) \ldots \ldots\left(1-x^n\right)\right\}^2}=\) _____________, \(\forall n \in N\)
MCQ+1 / -02021
47Limits Continuity And Differentiability
If \(\lim _\limits{x \rightarrow 0}\left(\frac{11 x^3-3 x+4}{13 x^3-5 x^2-7}\right)=\frac{a}{b}\), then the value of \(a+b\) equals
MCQ+1 / -02021
48Limits Continuity And Differentiability
If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then $$f(x)=\left\{\begin{array}{cc}x+a \sqrt{2}(\sin x) & , \quad 0 \leq x < \frac{\pi}{4} \\ 2 x(\cot x)+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a...
MCQ+1 / -02021
49Limits Continuity And Differentiability
$$f(x)=\left\{\begin{array}{cc}
\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\
K \log 2 \log 3, & x=0
\end{array}\right.$$
Find the value of \(k\) for which the function \(f\) is continuous.
Find the value of \(k\) for which the function \(f\) is continuous.
MCQ+1 / -02021
50Limits Continuity And Differentiability
\(\lim _\limits{z \rightarrow 1} \frac{z^{(1 / 3)}-1}{z^{(1 / 6)}-1}\) is equal to
MCQ+1 / -02021