Limits, Continuity & Differentiability · JEE Main previous year questions

Let S be the set of all functions $f:[0,1]\to \mathrm{R}$, which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists a $c\in (0,1)$, depending on f, such that

1. A.$|f(c)-f(1)|<|f^{\prime }(c)|$
2. B.$|f(c)+f(1)|<(1+c)|f^{\prime }(c)|$
3. C.$|f(c)-f(1)|<(1-c)|f^{\prime }(c)|$
4. D.None

${\lim }_{x\to 0}\frac{{\int }_0^{x}t\sin (10t)dt}{x}$ is equal to

1. A.$-\frac{1}{5}$
2. B.$-\frac{1}{10}$
3. C.0
4. D.$\frac{1}{10}$

Let the function, $f:[-7,0]\to \mathbb{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }(x)\le 2$, for all $x\in (-7,0)$, then for all such functions f, $f(-1)+f(0)$ lies in the interval:

1. A.$[-6,20]$
2. B.$(-\infty ,20]$
3. C.$[-3,11]$
4. D.$(-\infty ,11]$

${\lim }_{x\to 2}\frac{3^x+3^{3-x}-12}{3^{-x/2}-3^{1-x}}$ is equal to_____.

If the function f defined on $\left(-\frac{1}{3},\frac{1}{3}\right)$ by $$f(x)=\{\begin{array}{ll}\frac{1}{x}{\log }_e\left(\frac{1+3x}{1-2x}\right),& \text{when }x\ne 0\\ k,& \text{when }x=0\end{array}$$ is continuous, then k is equal to_______.

If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0$, then $\underset{n\to \infty }{\lim }\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{\lim }\sum _{r=1}^n{\beta }^r$ is equal to :

1. A.$\frac{7}{116}$
2. B.$\frac{29}{358}$
3. C.$\frac{1}{12}$
4. D.$\frac{21}{346}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function such that $f(2)=6$ and $f^{\prime }(2)=\frac{1}{48}$. If ${\int }_6^{f(x)}4t^{3}dt=(x-2)g(x)$, then ${\lim }_{x\to 2}g(x)$ is equal to :

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

${\lim }_{x\to 0}\frac{x+2\sin x}{\sqrt{x^2+2\sin x+1}-\sqrt{{\sin }^{2}x-x+1}}$ is :

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

Let $f(x)=5-|x-2|$ and $g(x)=|x+1|$, $x\in \mathbb{R}$. If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta$, then ${\lim }_{x\to -\alpha \beta }\frac{(x-1)(x^2-5x+6)}{x^2-6x+8}$ is equal to :

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

If $f(x)=\{\begin{array}{ll}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$ is continuous at x = 0, then the ordered pair (p, q) is equal to

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

Let $f:\mathrm{R}\to \mathrm{R}$ be differentiable at $c\in \mathrm{R}$ and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is :

1. A.differentiable if $f^{\prime }(c)=0$
2. B.differentiable if $f^{\prime }(c)\ne 0$
3. C.not differentiable
4. D.not differentiable if $f^{\prime }(c)=0$

${\lim }_{n\to \infty }\left(\frac{(n+1{)}^{1/3}}{n^{4/3}}+\frac{(n+2{)}^{1/3}}{n^{4/3}}+\cdots +\frac{(2n{)}^{1/3}}{n^{4/3}}\right)$ is equal to :

1. A.$\frac{4}{3}(2{)}^{3/4}$
2. B.$\frac{3}{4}(2{)}^{4/3}-\frac{3}{4}$
3. C.$\frac{4}{3}(2{)}^{4/3}$
4. D.$\frac{3}{4}(2{)}^{4/3}-\frac{4}{3}$

If ${\lim }_{x\to 1}\frac{x^2-ax+b}{x-1}=5$, then a + b is equal to :

1. A.1
2. B.- 4
3. C.- 7
4. D.5

If the function f defined on, $\left(\frac{\pi }{6},\frac{\pi }{3}\right)$ by $$f(x)=\{\begin{array}{ll}\frac{\sqrt{2}\cos x-1}{\cot x-1},& x\ne \frac{\pi }{4}\\ k,& x=\frac{\pi }{4}\end{array}$$ is continuous, then k is equal to

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

If $f:R\to R$ is a differentiable function and f(2) = 6, then $\underset{x\to 2}{\lim }\frac{{\int }_6^{f(x)}2tdt}{(x-2)}$ is :-

1. A.2f'(2)
2. B.24f'(2)
3. C.0
4. D.12f'(2)

If the function $f(x)=\{\begin{array}{l}a|\pi -x|+1,x\le 5\\ b|x-\pi |+3,x>5\end{array}$ is continuous at $x=5$, then the value of $a-b$ is :-

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

Let $f:R\to R$ be a differentiable function satisfying $f^{\prime }(3)+f^{\prime }(2)=0$. Then $\underset{x\to 0}{\lim }{\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)}^{\frac{1}{x}}$ is equal to

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

Let $f:[-1,3]\to R$ be defined as $$f(x)=\{\begin{array}{ll}|x|+[x],& -1\le x<1\\ x+|x|,& 1\le x<2\\ x+[x],& 2\le x\le 3\end{array}$$ where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at:

1. A.only three points
2. B.four or more points
3. C.only two points
4. D.only one point

Let f and g be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then ${\int }_0^{a}f(x)g(x)dx$ is equal to :

1. A.$4{\int }_0^{a}f(x)dx$
2. B.$-3{\int }_0^{a}f(x)dx$
3. C.${\int }_0^{a}f(x)dx$
4. D.$2{\int }_0^{a}f(x)dx$

${\lim }_{x\to \pi /4}\frac{{\cot }^{3}x-\tan x}{\cos (x+\frac{\pi }{4})}$ is :

1. A.$8\sqrt{2}$
2. B.4
3. C.$4\sqrt{2}$
4. D.8

${\lim }_{x\to 1^{-}}\frac{\sqrt{\pi }-\sqrt{2{\sin }^{-1}x}}{\sqrt{1-x}}$ is equal to :

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

${\lim }_{n\to \infty }\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\cdots +\frac{1}{5n}\right)$ is equal to :

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

${\lim }_{x\to 0}\frac{x\cot (4x)}{{\sin }^{2}x{\cot }^2(2x)}$ is equal to :

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

$$\underset{y\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$$

1. A.exists and equals $\frac{1}{2\sqrt{2}}$
2. B.exists and equals $\frac{1}{4\sqrt{2}}$
3. C.exists and equals $\frac{1}{2\sqrt{2}(1+\sqrt{2})}$
4. D.does not exist

${\lim }_{x\to 0}\frac{(27+x{)}^{\frac{1}{3}}-3}{9-(27+x{)}^{\frac{2}{3}}}$ equals.

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