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Questions >> JEEMAIN and NEET >> Mathematics >> Limit, Continuity and Differentiability

$f$ is defined in [-5,5] as $f(x)=\left\{\begin{array}{1 1}x&if\;x\;is\;rational\\-x&if\;x\;is\;irrational\end{array}\right.$ . Then

asked Jan 3, 2014 by sreemathi.v

1 answer

If $\lim\limits_{x\to \infty}\bigg[\large\frac{x^2+x+1}{x+1}$ $-ax-b\bigg]=4$ then

asked Jan 3, 2014 by sreemathi.v

1 answer

The value of $\lim\limits_{x\to 0}\big((\sin x)^{1/x}+(1+x)^{\large\sin x}\big)$ where $x > 0$ is

asked Jan 3, 2014 by sreemathi.v

1 answer

The function given by $y=||x|-1|$ is differentiable for all real numbers except the points

asked Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{n\to \infty}\large\frac{1}{n}$ $\sum\limits_{r=1}^{2n}\large\frac{r}{\sqrt{n^2+r^2}}$ equals

asked Jan 3, 2014 by sreemathi.v

1 answer

The function $f(x)=[x]\cos\big[\large\frac{2x-1}{2}\big]$ $\pi$ where [.] denotes the greatest integer function, is discontinuous at

asked Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{n\to \infty}\left\{\large\frac{1}{1-n^2} + \frac{1}{1-n^2} +......+ \large\frac{n}{1-n^2}\right\}$ is equal to

asked Jan 3, 2014 by sreemathi.v

1 answer

The function $f(x)=\large\frac{ln(1+ax)-ln(1-bx)}{x}$ is not defined at $x=0$ . The value which should be assigned to $f$ at $x=0$ so that it is continuous at $x=0$ is

asked Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}(\log_22x)^{\large\log_x5}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{(\cos x)^{1/2}-(\cos x)^{1/3}}{\sin^2x}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

If $a,b,c,d$ are positive then $\lim\limits_{x\to \infty}\big(1+\large\frac{1}{a+bx}\big)^{c+dx}=$

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f(x)=[x^2]-[x]^2$ where [.] is the largest integer function then

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f(x)$ is the integral function of the function $\large\frac{2\sin x-\sin 2x}{x^3}\qquad$ $x\neq 0$ then $\lim\limits_{x\to 0} f'(x)$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f(x)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$ then value of $\lim\limits_{\alpha \to 0}\large\frac{f(1-\alpha)-f(1)}{\alpha^3+3\alpha}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$f(x)=\sin^{-1}\big(\large\frac{2x}{1+x^2}\big)$ is differentiable on

asked Jan 2, 2014 by sreemathi.v

1 answer

If $\big(\large\frac{2+\sin x}{1+y}\big)\frac{dy}{dx}$ $=-\cos x,y(0)=1$ then $y\big(\large\frac{\pi}{2}\big)=$

asked Jan 2, 2014 by sreemathi.v

1 answer

If $\lim\limits_{x\to 0}\large\frac{((a-n)nx-\tan x)\sin nx}{x^2}$ $=0$ where $n$ is a non zero real number, then $a$ is equal to

asked Jan 2, 2014 by sreemathi.v

1 answer

Which of the following function is differentiable at $x=0$ ?

asked Jan 2, 2014 by sreemathi.v

1 answer

Let $f:(0,\infty)\rightarrow R$ and $f(x)=\int_0^xf(t) dt$ if $f(x^2)=x^2(1+x)$ then f(4) equals

asked Jan 2, 2014 by sreemathi.v

1 answer

Suppose $f:R\rightarrow R$ is a differentiable function and $f(1)=4$ .Then value of $\lim\limits_{x\to 1}\int\limits_4^{f(x)}\large\frac{2t}{x-1}$ $dt$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\big(\large\frac{x^2+5x+3}{x^2+x+2}\big)^x$ =

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{n\to \infty}\bigg[\large\frac{1}{1-n^2}+\frac{2}{1-n^2}+..........+\frac{n}{1-n^2}\bigg]$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{2^x-1}{\sqrt{1+x}-1}$ =

asked Jan 2, 2014 by sreemathi.v

1 answer

In order that the function $f(x)=(x+1)^{\large\cot x}$ is continuous at $x=0$ . f(0) must be defined as

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}\big[\sec\big(\large\frac{\pi x}{2}\big)$ $\log x\bigg]$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f$ is twice differentiable and $f''(0)=2$ then $\lim\limits_{x\to 0}\large\frac{2f(x)-3f(2x)+f(4x)}{x^2}$ =

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{(1+x)^{1/x}-e}{x}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0^+}x^m(\log x)^n,(m,n\in N)$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

The value of $\lim\limits_{n\to \infty}x\bigg[\tan^{-1}\large\frac{x+1}{x+2}$ $-\cot^{-1}\large\frac{x+2}{x}\bigg]$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f(x)=\left\{\begin{array}{1 1}\large\frac{1-\cos\lambda x}{x\sin x}&x\neq 0\\\large\frac{1}{2}&x=0\end{array}\right.$ is continuous at $x=0$ then $\lambda$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

Find the value of p for which the function $f(x)=\left\{\begin{array}{ 1 1}\large\frac{(4^x-1)^3}{\sin\big(x/p\big)\log \big(1+\Large\frac{x^2}{3}\big)}&x\neq 0\\12(\log 4)^3&x=0\end{array}\right.$ is continuous at $x=0$

asked Dec 31, 2013 by sreemathi.v

1 answer

Find the points of discontinuity of $y=\large\frac{1}{u^2+u-2}$ where $u=\large\frac{1}{x-1}$

asked Dec 31, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{\sin x}{x}$

asked Dec 31, 2013 by sreemathi.v

1 answer

Find $\lim\limits_{x\to 0}\{\tan(\large\frac{\pi}{4}$ $+x)\}^{1/x}$

asked Dec 31, 2013 by sreemathi.v

1 answer

Use the formula $\lim\limits_{x\to 0}\large\frac{a^x-1}{x}=$ $ln\; a$ to find $\lim\limits_{x\to 0}\large\frac{2^x-1}{(1+x)^{1/2}-1}$

asked Dec 31, 2013 by sreemathi.v

1 answer

$f(x)$ is the integral of $\large\frac{2\sin x-\sin 2x}{x^3}$ $x\neq 0$ find $\lim\limits_{x\to 0}f'(x)$

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f:R\rightarrow R$ be a function such that $f(x+y)=f(x)+f(y)\forall x,y\in R$ if $f(x)$ is differentiable at $x=0$ then

asked Dec 31, 2013 by sreemathi.v

1 answer

The values of $p$ and $q$ for which the function $f(x)=\left\{\begin{array}{1 1}\large\frac{\sin(p+1)x+\sin x}{x} & x < 0 \\ q & x=0 \\ \large\frac{\sqrt{x+x^2}-\sqrt x}{x^3/2} & x > 0 \end{array}\right.$ is continuous for all $x\;in\;R$ are

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f:R\rightarrow R$ be a function defined by $f(x)=min\{x+1,\mid x\mid+1\}$ . Then which of the following is true ?

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f$ is a real valued differentiable function satisfying $\mid f(x)-f(y)\mid\leq (x-y)^2,x,y\in R$ and $f(0)=0$ then $f(1)$ equals

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f(a)=g(a)=k$ and their $n^{th}$ derivatives $f^n(a)$ , $g^n(a)$ exist and are not equal for some n. Further if $\lim\limits_{x\to a}\large\frac{f(a)g(x)-f(a)-g(a)f(x)+f(a)}{g(x)-f(x)}$ $=4$ , then the value of k is

asked Dec 31, 2013 by sreemathi.v

1 answer

$f(x)$ and $g(x)$ are two differentiable function on [0,2] such that $f''(x)-g''(x)=0$ . $f'(1)=2g'(1)=4f(2)=3g(2)=9$ then $f(x)-g(x)$ at $x=\large\frac{3}{2}$ is

asked Dec 31, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}\big[\large\frac{x^2+5x+3}{x^2+x+3}\big]^x$

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f(x)$ is continuous and differentiable function and $f(1/n)=0\forall n \geq 1$ and $n\in 1$ then

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f:R\rightarrow R$ be such that $f(1)=3$ and $f'(1)=6$ . Then $\lim\limits_{x\to 0}\big[\large\frac{f(1+x)}{f(1)}\big]^{\Large\frac{1}{x}}$ equals

asked Dec 31, 2013 by sreemathi.v

1 answer

The left hand derivative of $f(x)=[x]\sin(\pi x)$ at $x=k$ , where $k$ is an integer is

asked Dec 31, 2013 by sreemathi.v

1 answer

Let [.] denote the greatest integer function and $f(x)=[\tan^2x]$ then

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f:R\rightarrow R$ be a differentiable function and $f(1)=4$ , then the value of $\large \int_4^{f(x)}\large\frac{2t\;dt}{x-1}$ is

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f(x)=\left\{\begin{array}{1 1}\large\frac{\sin [x]}{[x]}&[x]\neq 0\\0&[x]=0\end{array}\right.$ , where $[x]$ denotes the greatest integer less than or equal to x, then $\lim\limits_{x\to 0}f(x)$ equals

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f(a)=2$ , $f'(a)=1$ , $g(a)=-1$ , $g'(a)=2$ , then the value of $\lim\limits_{x\to a}\large\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

asked Dec 31, 2013 by sreemathi.v

1 answer

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