Limits, Continuity and Differentiability · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1

Let $f(x)=\left\{\begin{array}{cl}1+\frac{2 x}{a}, & 0 \leq x \leq 1 \\ a x, & 1 < x \leq 2\end{array}\right.$.If $\lim _{x \rightarrow 1} f(x)$ exists, then the sum of the cubes of the possible values of $a$ is

AP EAPCET 2024 - 23th May Morning Shift
2

Let $[P]$ denote the greatest integer $\leq P$. If $0 \leq a \leq 2$, then the number of integral values of ' $a$ ' such that $\lim \limits_{x \rightarrow a}\left(\left[x^2\right]-[x]^2\right)$ does not exist is

AP EAPCET 2024 - 23th May Morning Shift
3
If $f(x)=\left\{\begin{array}{cl}\frac{\sqrt{a^2-a x+x^2}-\sqrt{x^2+a x+a^2}}{\sqrt{a+x}-\sqrt{a-x}}, & x \neq 0 \text { is } \\ K & x=0\end{array}\right.$ continuous at $x=0$, then $K$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
4
If $f(x)=\left\{\begin{array}{cc}a x^2+b x-\frac{13}{8}, & x \leq 1 \\ 3 x-3, & 1 < x \leq 2 \text { is differentiable } \\ b x^3+1, & x > 2\end{array}\right.$ $\forall x \in R$, then $a-b$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
5
In each of the following options, a function and an interval are given. Choose the option containing the function and the interval for which Lagrange's mean value theorem is not applicable
AP EAPCET 2024 - 23th May Morning Shift
6
The function $f(x)=\left\{\begin{array}{cc}\frac{x-|x|}{x}, & x \neq 0 \\ 2, & x=0\end{array}\right.$
AP EAPCET 2024 - 23th May Morning Shift
7

$$\mathop {\lim }\limits_{x \to \infty } \frac{[2 x-3]}{x} \text { is equal to } $$

AP EAPCET 2024 - 22th May Evening Shift
8
$\mathop {\lim }\limits_{x \to 0}\frac{\cos 2 x-\cos 3 x}{4 x-\cos 5 x}$ is equal to $\cos 4 x-\cos 5 x$
AP EAPCET 2024 - 22th May Evening Shift
9

If a real valued function $f(x)=\left\{\begin{array}{cl}\frac{2 x^2+(k+2) x+9}{3 x^2-7 x-6}, & \text { for } x \neq 3 \\ 1, & \text { for } x=3\end{array}\right.$ is continuous at $x=3$ and $l$ is a finite value, then $l-k$ is equal to

AP EAPCET 2024 - 22th May Evening Shift
10

$$\mathop {\lim }\limits_{x \to o} \left[\frac{1}{x}-\frac{1}{e^x-1}\right]= $$

AP EAPCET 2024 - 22th May Morning Shift
11

Let $f(x)=\left\{\begin{array}{cl}0, & x=0 \\ 2-x, & \text { for } 0 < x < 1 \\ 2, & \text { for } x=1 \\ \frac{1}{2}-x, & \text { for } 1 < x < 2 \\ \frac{-3}{2}, & \text { for } x \geq 2\end{array}\right.$

then which of the following is true

AP EAPCET 2024 - 22th May Morning Shift
12
If $f(x)=\left(\frac{1+x}{1-x}\right)^{\frac{1}{x}}$ is continuous at $x=0$, then $f(0)=$
AP EAPCET 2024 - 22th May Morning Shift
13
The function $f(x)=|x-24|$ is
AP EAPCET 2024 - 22th May Morning Shift
14
$$\mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2-1}}+\ldots+\frac{1}{\sqrt{n^2-(n-1)^2}}\right)= $$
AP EAPCET 2024 - 22th May Morning Shift
15
$$\mathop {\lim }\limits_{x \to 0} \left( {{{\sin (\pi {{\cos }^2}x} \over {{x^2}}}} \right) = $$
AP EAPCET 2024 - 21th May Evening Shift
16
$$\mathop {\lim }\limits_{x \to 1} \left( {{{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}} \right) = $$
AP EAPCET 2024 - 21th May Evening Shift
17
If the function $f(x)=\frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$, then $f(0)=$
AP EAPCET 2024 - 21th May Evening Shift
18
If $f(x)=\frac{5 x \cdot \operatorname{cosec}(\sqrt{x})-1}{(x-2) \operatorname{cosec}(\sqrt{x})}$, then $\lim \limits_{x \rightarrow \infty} f\left(x^2\right)=$
AP EAPCET 2024 - 21th May Morning Shift
19
$\lim \limits_{x \rightarrow 2} \frac{\sqrt{1+4 x}-\sqrt{3+3 x}}{x^3-8}=$
AP EAPCET 2024 - 21th May Morning Shift
20
If $$ \lim _{x \rightarrow \infty} \frac{(\sqrt{2 x+1}+\sqrt{2 x-1})^8+(\sqrt{2 x+1}-\sqrt{2 x-1})^8\left(P x^4-16\right)}{\left(x+\sqrt{x^2-2}\right)^8+\left(x-\sqrt{x^2-2}\right)^8}=1 $$ then $P=$
AP EAPCET 2024 - 21th May Morning Shift
21
$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x}=$
AP EAPCET 2024 - 20th May Evening Shift
22
If $\lim \limits_{x \rightarrow 0} \frac{e^x-a-\log (1+x)}{\sin x}=0$, then $a=$
AP EAPCET 2024 - 20th May Evening Shift
23

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{\pi}{6}\end{array}\right. $

are

AP EAPCET 2024 - 20th May Evening Shift
24

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 $f^{\prime}(2)=$

AP EAPCET 2024 - 20th May Evening Shift
25
In the interval $[0,3]$ The function $f(x)=|x-1|+|x-2|$ is
AP EAPCET 2024 - 20th May Evening Shift
26
$\lim \limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2+x^5+x^6}}{x^4}=$
AP EAPCET 2024 - 20th May Morning Shift
27
$\lim \limits_{x \rightarrow 1} \frac{\sqrt{x}-1}{\left(\cos ^{-1} x\right)^2}=$
AP EAPCET 2024 - 20th May Morning Shift
28

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 $|\alpha|+|\beta|=$

AP EAPCET 2024 - 20th May Morning Shift
29
$$ \lim \limits_{x \rightarrow 3} \frac{x^3-27}{x^2-9}= $$
AP EAPCET 2024 - 19th May Evening Shift
30

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

AP EAPCET 2024 - 19th May Evening Shift
31
The function $f(x)=\left\{\begin{array}{ll}\frac{2}{5-x}, & x<3 \\ 5-x, & x \geq 3\end{array}\right.$ is
AP EAPCET 2024 - 19th May Evening Shift
32

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?

AP EAPCET 2024 - 19th May Evening Shift
33
Let $f(x)=\min \left\{x, x^2\right\}$ for every real number of $x$, then
AP EAPCET 2024 - 19th May Evening Shift
34
$\lim \limits_{x \rightarrow 0} \frac{1-\cos x \cdot \cos 2 x}{\sin ^2 x}=$
AP EAPCET 2024 - 18th May Morning Shift
35
$\lim \limits_{x \rightarrow-1}\left(\frac{3 x^2-2 x+3}{3 x^2+x-2}\right)^{3 x-2}=$
AP EAPCET 2024 - 18th May Morning Shift
36

$f(x)=\left\{\begin{array}{cl}\frac{\left(2 x^2-a x+1\right)-\left(a x^2+3 b x+2\right)}{x+1}, & \text { if } x \neq-1 \\ k_k, & \text { if } x=-1\end{array}\right.$

is a real valued function. If $a, b, k \in R$ and $f$ is continuous on $R$, then $k=$

AP EAPCET 2024 - 18th May Morning Shift
37
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
AP EAPCET 2024 - 18th May Morning Shift
38

$$\lim _\limits{x \rightarrow-\infty} \log _e(\cosh x)+x=$$

AP EAPCET 2022 - 5th July Morning Shift
39

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 $$a+2 c=$$

AP EAPCET 2022 - 5th July Morning Shift
40

$$\lim _\limits{x \rightarrow-\infty} \frac{3|x|-x}{|x|-2 x}-\lim _\limits{x \rightarrow 0} \frac{\log \left(1+x^3\right)}{\sin ^3 x}=$$

AP EAPCET 2022 - 5th July Morning Shift
41

If $$[\cdot]$$ denotes greatest integer function, then $$\lim _\limits{x \rightarrow \frac{-3}{5}} \frac{1}{\dot{x}}\left[\frac{-1}{x}\right]=$$

AP EAPCET 2022 - 4th July Evening Shift
42

If $$l, m(l< m)$$ are roots of $$a x^2+b x+c=0$$, then $$\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=$$

AP EAPCET 2022 - 4th July Evening Shift
43

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 possible values for $$a$$ and $$b$$ are

AP EAPCET 2022 - 4th July Evening Shift
44

$$\frac{d}{d x}\left(\lim _{x \rightarrow 2} \frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)=$$

AP EAPCET 2022 - 4th July Evening Shift
45

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

AP EAPCET 2022 - 4th July Evening Shift
46

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, $$\lim _\limits{x \rightarrow 0} \frac{(f(x))^{1 / 3}-1}{(f(x))^{1 / 2}-1}=$$

AP EAPCET 2022 - 4th July Morning Shift
47

$$\begin{aligned} & \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\ & \lim _{x \rightarrow 0} x^4 \sin \left(\frac{1}{3 \sqrt{x}}\right)=l \text {. Then, } k+l= \end{aligned}$$

AP EAPCET 2022 - 4th July Morning Shift
48

If $$\lim _\limits{n \rightarrow \infty} x^n \log _e x=0$$, then $$\log _x 12=$$

AP EAPCET 2022 - 4th July Morning Shift
49

If $$f(x)=\operatorname{Max}\{3-x, 3+x, 6\}$$ is not differentiable at $$x=a$$, and $$x=b$$, then $$|a|+|b|=$$

AP EAPCET 2022 - 4th July Morning Shift
50

$$\lim _\limits{n \rightarrow \infty}\left(\frac{1}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)=$$

AP EAPCET 2022 - 4th July Morning Shift
51

$$\mathop {\lim }\limits_{n \to \infty } {{n{{(2n + 1)}^2}} \over {(n + 2)({n^2} + 3n - 1)}}$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
52

If the function $$f(x)$$, defined below, is continuous on the interval $$[0,8]$$, then $$f(x)=\left\{\begin{array}{cc}x^2+a x+b & , \quad 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b & , 4 < x \leq 8\end{array}\right.$$

AP EAPCET 2021 - 20th August Morning Shift
53

If $$f(x)$$, defined below, is continuous at $$x=4$$, then

$$f(x) = \left\{ {\matrix{ {{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr {a + b} & , & {x = 4} \cr {{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr } } \right.$$

AP EAPCET 2021 - 20th August Morning Shift
54

If $$f(x)=\left\{\begin{array}{cc}\frac{e^{\alpha x}-e^x-x}{x^2}, & x \neq 0 \\ \frac{3}{2}, & x=0\end{array}\right.$$

Find the value of $$\alpha$$ for which the function $$f$$ is continuous

AP EAPCET 2021 - 19th August Evening Shift
55

The value of $$k(k > 0)$$, for which the function $$f(x)=\frac{\left(e^x-1\right)^4}{\sin \left(\frac{x^2}{k^2}\right) \log \left(1+\frac{x^2}{2}\right)}$$, where $$x \neq 0$$ and $$f(0)=8$$

AP EAPCET 2021 - 19th August Evening Shift
56

If $$f^{\prime \prime}(x)$$ is continuous at $$x=0$$ and $$f^{\prime \prime}(0)=4$$, then find the following value. $$\lim _\limits{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^2}$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
57

$$\lim _\limits{z \rightarrow 1} \frac{z^{(1 / 3)}-1}{z^{(1 / 6)}-1}$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
58

$$f(x)=\left\{\begin{array}{cc} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ K \log 2 \log 3, & x=0 \end{array}\right.$$

Find the value of $$k$$ for which the function $$f$$ is continuous.

AP EAPCET 2021 - 19th August Morning Shift
59

If the function $$f(x)$$, defined below is continuous in the interval $$[0, \pi]$$, then $$f(x)=\left\{\begin{array}{cc}x+a \sqrt{2}(\sin x) & , \quad 0 \leq x < \frac{\pi}{4} \\ 2 x(\cot x)+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), & \frac{\pi}{2} < x \leq \pi\end{array}\right.$$

AP EAPCET 2021 - 19th August Morning Shift
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