Limits, Continuity & Differentiability · JEE Main previous year questions

If ${\lim }_{x\to 0}\frac{{\sin }^{-1}x-{\tan }^{-1}x}{3x^3}$ is equal to $L$, then the value of $(6L+1)$ is

1. A.$\frac{1}{6}$
2. B.$\frac{1}{2}$
3. C.6
4. D.2

If $f(x)=\{\begin{array}{ll}\frac{1}{|x|}& ;|x|\ge 1\\ a{x}^2+b& ;|x|<1\end{array}$ is differentiable at every point of the domain, then the values of a and b are respectively :

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

Let $f:\mathrm{R}\to \mathrm{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}\frac{\sin (a+1)x+\sin 2x}{2x},& \text{if }x<0\\ b,& \text{if }x=0\\ \frac{\sqrt{x+b{x}^3}-\sqrt{x}}{b{x}^{5/2}},& \text{if }x>0\end{array}$$ If f is continuous at x = 0, then the value of a + b is equal to :

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

Let $g(x)={\int }_0^{x}f(t)dt$, where f is continuous function in [ 0, 3 ] such that $\frac{1}{3}\le f(t)\le 1$ for all $t\in [0,1]$ and $0\le f(t)\le \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which g(3) lies is :

1. A.$[-1,-\frac{1}{2}]$
2. B.$[-\frac{3}{2},-1]$
3. C.$[1,3]$
4. D.$[\frac{1}{3},2]$

Let $f:\mathrm{R}\to \mathrm{R}$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathrm{R}$ and $f(x)\ne 0$ for any $x\in \mathrm{R}$. If the function f is differentiable at $x=0$ and $f^{\prime }(0)=3$, then ${\lim }_{h\to 0}\frac{1}{h}(f(h)-1)$ is equal to ____.

Let $S_k={\sum }_{r=1}^k{\tan }^{-1}\left(\frac{6^r}{2^{2r+1}+3^{2r+1}}\right)$. Then ${\lim }_{k\to \infty }{S}_k$ is equal to :

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

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in \mathbb{R}$. If $I_1={\int }_0^{8}f(x)dx$ and $I_2={\int }_{-1}^{3}f(x)dx$, then the value of $I_1+2I_2$ is equal to _________.

If ${\lim }_{x\to 0}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}=2$, then $a+b+c$ is equal to _________.

Let $f:(0,2)\to \mathbb{R}$ be defined as $f(x)={\log }_2\left(1+\tan \left(\frac{\pi x}{4}\right)\right)$. Then, ${\lim }_{n\to \infty }\frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots +f(1)\right)$ is equal to _________.

The value of ${\lim }_{h\to 0}2\left\{\frac{\sqrt{3}\sin \left(\frac{\pi }{6}+h\right)-\cos \left(\frac{\pi }{6}+h\right)}{\sqrt{3}h\left(\sqrt{3}\cos h-\sin h\right)}\right\}$ is :

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

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then ${\lim }_{x\to a}\frac{xf(a)-af(x)}{x-a}$ equals :

1. A.$4-2a$
2. B.$2a+4$
3. C.$a+4$
4. D.$2a-4$

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=\{\begin{array}{ll}2\sin \left(-\frac{\pi x}{2}\right),& \text{if }x<-1\\ |a{x}^2+x+b|,& \text{if }-1\le x\le 1\\ \sin (\pi x),& \text{if }x>1\end{array}$ If f(x) is continuous on $\mathbb{R}$, then $a+b$ equals :

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

$$\underset{n\to \infty }{\lim }\left[\frac{1}{n}+\frac{n}{(n+1{)}^2}+\frac{n}{(n+2{)}^2}+\cdots +\frac{n}{(2n-1{)}^2}\right]\text{ is equal to :}$$

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

If ${\lim }_{x\to 0}\frac{ax-(e^{4x}-1)}{ax(e^{4x}-1)}$ exists and is equal to b, then the value of $a-2b$ is ______ .

$$\underset{x\to 1}{\lim }\left(\frac{{\int }_0^{(x-1{)}^2}t\cos (t^2)dt}{(x-1)\sin (x-1)}\right)$$

1. A.is equal to 0
2. B.is equal to $\frac{1}{2}$
3. C.does not exist
4. D.is equal to $-\frac{1}{2}$

Let $[t]$ denote the greatest integer $\le t$. If for some $\lambda \in \mathbb{R}-\{1,0\}$, ${\lim }_{x\to 0}\left|\frac{1-x+|x|}{\lambda -x+[x]}\right|=\mathrm{L}$, then L is equal to :

1. A.1
2. B.2
3. C.0
4. D.$\frac{1}{2}$

If ${\lim }_{x\to 0}\left\{\frac{1}{x^8}\left(1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2}\cos \frac{x^2}{4}\right)\right\}=2^{-k}$ then the value of k is _______.

${\lim }_{x\to a}\frac{(a+2x{)}^{\frac{1}{3}}-(3x{)}^{\frac{1}{3}}}{(3a+x{)}^{\frac{1}{3}}-(4x{)}^{\frac{1}{3}}}(a\ne 0)$ is equal to :

1. A.$\left(\frac{2}{9}\right){\left(\frac{2}{3}\right)}^{\frac{1}{3}}$
2. B.$\left(\frac{2}{3}\right){\left(\frac{2}{9}\right)}^{\frac{1}{3}}$
3. C.${\left(\frac{2}{3}\right)}^{\frac{4}{3}}$
4. D.${\left(\frac{2}{9}\right)}^{\frac{4}{3}}$

If a function f(x) defined by $$f(x)=\{\begin{array}{ll}a{e}^x+b{e}^{-x},& -1\le x<1\\ c{x}^2,& 1\le x\le 3\\ a{x}^2+2cx,& 3<x\le 4\end{array}$$ be continuous for some $a,b,c\in \mathbb{R}$ and $f^{\prime }(0)+f^{\prime }(2)=e$, then the value of $a$ is :

1. A.$\frac{e}{e^2-3e-13}$
2. B.$\frac{1}{e^2-3e+13}$
3. C.$\frac{e}{e^2-3e+13}$
4. D.$\frac{e}{e^2+3e+13}$

${\lim }_{x\to 0}{\left(\tan \left(\frac{\pi }{4}+x\right)\right)}^{\frac{1}{x}}$ is equal to :

1. A.2
2. B.1
3. C.$e$
4. D.$e^2$

Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$. If for all $x\in (a,b)$, $f^{\prime }(x)>0$ and $f^{\prime \prime }(x)<0$, then for any $c\in (a,b)$, $\frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than :

1. A.1
2. B.$\frac{b-c}{c-a}$
3. C.$\frac{b+a}{b-a}$
4. D.$\frac{c-a}{b-c}$

If $$f(x)=\{\begin{array}{llc}\frac{\sin (a+2)x+\sin x}{x}& ;& x<0\\ b& ;& x=0\\ \frac{(x+3x^2{)}^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}& ;& x>0\end{array}$$ is continuous at $x=0$, then $a+2b$ is equal to :

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

Let $[t]$ denote the greatest integer $\le t$ and ${\lim }_{x\to 0}x\left[\frac{4}{x}\right]=A$. Then the function, $f(x)=[x^2]\sin (\pi x)$ is discontinuous, when x is equal to :

1. A.$\sqrt{A+1}$
2. B.$\sqrt{A+5}$
3. C.$\sqrt{A+21}$
4. D.$\sqrt{A}$

Let a function $f:[0,5]\to \mathbb{R}$ be continuous, $f(1)=3$ and F be defined as : $$F(x)={\int }_1^x{t}^{2}g(t)dt\text{ , where }g(t)={\int }_1^{t}f(u)du$$ Then for the function F, the point $x=1$ is :

1. A.a point of inflection.
2. B.a point of local maxima.
3. C.a point of local minima.
4. D.not a critical point.

${\lim }_{x\to 0}{\left(\frac{3x^2+2}{7x^2+2}\right)}^{\frac{1}{x^2}}$ is equal to

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