Limits, Continuity & Differentiability · JEE Main previous year questions

If the function f defined as $f(x)=\frac{1}{x}-\frac{k-1}{e^{2x}-1},x\ne 0$, is continuous at $x=0$, then the ordered pair (k, f(0)) is equal to :

1. A.(3, 2)
2. B.(3, 1)
3. C.(2, 1)
4. D.$\left(\frac{1}{3},2\right)$

If $f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$, then ${\lim }_{x\to 0}\frac{f^{\prime }(x)}{x}$

1. A.does not exist.
2. B.exists and is equal to 2.
3. C.exists and is equal to 0.
4. D.exists and is equal to $-2$.

For each $t\in R$, let [t] be the greatest integer less than or equal to t. Then ${\lim }_{x\to 0^{+}}x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\dots \cdots +\left[\frac{15}{x}\right]\right)$

1. A.does not exist in R
2. B.is equal to 0
3. C.is equal to 15
4. D.is equal to 120

Let $\mathrm{S}=\{t\in R:f(x)=|x-\pi |\cdot (e^{|x|}-1)\sin |x|\text{ is not differentiable at t}\}$, then the set S is equal to

1. A.$\{0,\pi \}$
2. B.$\varphi$ (an empty set)
3. C.$\{0\}$
4. D.$\{\pi \}$

The value of k for which the function $$f(x)=\{\begin{array}{ll}{\left(\frac{4}{5}\right)}^{\frac{\tan 4x}{\tan 5x}},& 0<x<\frac{\pi }{2}\\ k+\frac{2}{5},& x=\frac{\pi }{2}\end{array}$$ is continuous at $x=\frac{\pi }{2}$, is :

1. A.$\frac{17}{20}$
2. B.$\frac{2}{5}$
3. C.$\frac{3}{5}$
4. D.$-\frac{2}{5}$

If ${\lim }_{n\to \infty }\frac{1^a+2^a+\dots \cdots +n^a}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots \cdots +(na+n)]}=\frac{1}{60}$ for some positive real number a, then a is equal to :

1. A.$7$
2. B.$8$
3. C.$\frac{15}{2}$
4. D.$\frac{17}{2}$

${\lim }_{x\to 3}\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}$ is equal to :

1. A.$\sqrt{3}$
2. B.$\frac{1}{\sqrt{2}}$
3. C.$\frac{\sqrt{3}}{2}$
4. D.$\frac{1}{2\sqrt{2}}$

Let $p={\lim }_{x\to 0^{+}}{\left(1+{\tan }^2\sqrt{x}\right)}^{\frac{1}{2x}}$ then $\log p$ is equal to :

1. A.$\frac{1}{2}$
2. B.$\frac{1}{4}$
3. C.$2$
4. D.$1$

${\lim }_{n\to \infty }{\left(\frac{(n+1)(n+2)\dots 3n}{n^{2n}}\right)}^{\frac{1}{n}}$ is equal to:

1. A.$\frac{9}{e^2}$
2. B.$3\log 3-2$
3. C.$\frac{18}{e^4}$
4. D.$\frac{27}{e^2}$

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If ${\lim }_{x\to 0}\left[1+\frac{f(x)}{x^2}\right]=3$, then $f(2)$ is equal to :

1. A.$0$
2. B.$4$
3. C.$-8$
4. D.$-4$

${\lim }_{x\to 0}\frac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$ is equal to

1. A.$2$
2. B.$\frac{1}{2}$
3. C.$4$
4. D.$3$

${\lim }_{x\to 0}\frac{\sin (\pi {\cos }^{2}x)}{x^2}$ is equal to :

1. A.$-\pi$
2. B.$\pi$
3. C.$\frac{\pi }{2}$
4. D.1

${\lim }_{x\to 0}\frac{\sin (\pi {\cos }^{2}x)}{x^2}$ equals

1. A.$-\pi$
2. B.1
3. C.$-1$
4. D.$\pi$

If $f(x)=a|\sin x|+b{e}^{|x|}+c|x{|}^3$, where $a,b,c\in R$, is differentiable at $x=0$, then

1. A.$a=0,b\text{ and }c\text{ are any real numbers}$
2. B.$c=0,a=0,b\text{ is any real number}$
3. C.$b=0,c=0,a\text{ is any real number}$
4. D.$a=0,b=0,c\text{ is any real number}$

Let $f:(-\infty ,\infty )\to (-\infty ,\infty )$ be defined by $f(x)=x^3+1$ Statement 1: The function $f$ has a local extremum at $x=0$ Statement 2: The function $f$ is continuous and differentiable on $(-\infty ,\infty )$ and $f^{\prime }(0)=0$

1. A.Statement 1 is true, Statement 2 is false.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
3. C.Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.
4. D.Statement 1 is false, Statement 2 is true.

Statement 1: A function $f:R\to R$ is continuous at $x_0$ if and only if ${\lim }_{x\to x_0}f(x)$ exists and ${\lim }_{x\to x_0}f(x)=f(x_0)$. Statement 2: A function $f:R\to R$ is discontinuous at $x_0$ if and only if, ${\lim }_{x\to x_0}f(x)$ exists and ${\lim }_{x\to x_0}f(x)\ne f(x_0)$.

1. A.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.
2. B.Statement 1 is false, Statement 2 is true.
3. C.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
4. D.Statement 1 is true, Statement 2 is false.

${\lim }_{x\to 0}\left(\frac{x-\sin x}{x}\right)\sin \left(\frac{1}{x}\right)$

1. A.equals 1
2. B.equals 0
3. C.does not exist
4. D.equals $-1$

Let $f:R\to R$ be a continuous function defined by $$f(x)=\frac{1}{e^x+2e^{-x}}$$ Statement - 1 : $f(c)=\frac{1}{3}$, for some $c\in R$. Statement - 2 : $0<f(x)\le \frac{1}{2\sqrt{2}}$, for all $x\in R$

1. A.Statement - 1 is true, Statement -2 is true; Statement - 2 is not a correct explanation for Statement - 1.
2. B.Statement - 1 is true, Statement - 2 is false.
3. C.Statement - 1 is false, Statement - 2 is true.
4. D.Statement - 1 is true, Statement -2 is true; Statement -2 is a correct explanation for Statement - 1.

Let $f:R\to R$ be a positive increasing function with $$\underset{x\to \infty }{\lim }\frac{f(3x)}{f(x)}=1\text{. Then }\underset{x\to \infty }{\lim }\frac{f(2x)}{f(x)}=$$

1. A.$\frac{2}{3}$
2. B.$\frac{3}{2}$
3. C.$3$
4. D.$1$

Let $f(x)=x|x|$ and $g(x)=\sin x$. Statement-1: gof is differentiable at $x=0$ and its derivative is continuous at that point. Statement-2: gof is twice differentiable at $x=0$.

1. A.Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
2. B.Statement-1 is true, Statement-2 is false
3. C.Statement-1 is false, Statement-2 is true
4. D.Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1

Let $f:R\to R$ be a differentiable function having $f(2)=6$, $f^{\prime }(2)=\left(\frac{1}{48}\right)$. Then ${\lim }_{x\to 2}{\int }_6^{f(x)}\frac{4t^3}{x-2}dt$ equals :

1. A.$24$
2. B.$36$
3. C.$12$
4. D.$18$

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\left(\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)$. Then $f\left(\frac{\pi }{2}\right)$ is

1. A.$\left(\frac{\pi }{4}+\sqrt{2}-1\right)$
2. B.$\left(\frac{\pi }{4}-\sqrt{2}+1\right)$
3. C.$\left(1-\frac{\pi }{4}-\sqrt{2}\right)$
4. D.$\left(1-\frac{\pi }{4}+\sqrt{2}\right)$

Let $\alpha$ and $\beta$ be the distinct roots of $a{x}^2+bx+c=0$, then $$\underset{x\to \alpha }{\lim }\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$$ is equal to

1. A.$\frac{a^2(\alpha -\beta {)}^2}{2}$
2. B.$0$
3. C.$-\frac{a^2(\alpha -\beta {)}^2}{2}$
4. D.$\frac{(\alpha -\beta {)}^2}{2}$

Suppose $f(x)$ is differentiable at $x=1$ and $$\underset{h\to 0}{\lim }\frac{1}{h}f(1+h)=5,$$ then $f^{\prime }(1)$ equals

1. A.$3$
2. B.$4$
3. C.$5$
4. D.$6$

$$\underset{n\to \infty }{\lim }\left[\frac{1}{n^2}{\sec }^2\frac{1}{n^2}+\frac{2}{n^2}{\sec }^2\frac{4}{n^2}\cdots +\frac{1}{n}{\sec }^{2}1\right]$$ equals

1. A.$\frac{1}{2}\sec 1$
2. B.$\frac{1}{2}\mathrm{cosec}1$
3. C.$\tan 1$
4. D.$\frac{1}{2}\tan 1$