Recent questions in Limit, Continuity and Differentiability

For a real number y, let [y] denotes the greatest integer less than or equal to y. Then the function $f(x)=\large\frac{\tan(\pi[x-\pi])}{1+[x]^2}$ is

asked Dec 31, 2013 by sreemathi.v

1 answer

Let $f(x)$ be a continuous function defined for $1\leq x\leq 3$ . If $f(x)$ takes rational value for all $x$ and $f(2) = 10$ then $f(1.5)$ =

asked Dec 31, 2013 by sreemathi.v

1 answer

$\lim\limits_{h\to 0}\large\frac{ln(1+2h)-2ln(1+h)}{h^2}=$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\bigg[\large\frac{1+5x^2}{1+3x^2}\bigg]^{\Large\frac{1}{x^2}}$

asked Dec 30, 2013 by sreemathi.v

1 answer

Let $f(x)=\left\{\begin{array}{1 1}\big[\tan(\large\frac{\pi}{4}\normalsize +x)\big]^{\large\frac{1}{x}}&x\neq 0\\k&x=0\end{array}\right.$ .For what value of k, $f(x)$ is continuous at $x=0$

asked Dec 30, 2013 by sreemathi.v

1 answer

If $f(x)=\large\frac{1}{1-x}$ ,find the points of discontinuity of the composite function $y=f(f(f(x)))$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to a}\large\frac{\log(x-a)}{\log(e^x-e^a)}$ is

asked Dec 30, 2013 by sreemathi.v

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If $f(x)=\large\frac{1}{\sqrt{18-x^2}}$ then $\lim\limits_{x\to 3}\large\frac{f(x)-f(3)}{x-3}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to {-1}}\large\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}}$ is given by

asked Dec 30, 2013 by sreemathi.v

1 answer

If $f(9)=9,f'(9)=4$ then $\lim\limits_{x\to 9}\large\frac{\sqrt{f(x)}-3}{\sqrt x-3}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\big(\large\frac{a^x+b^x+c^x}{3}\big)^{1/x}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \large\frac{\pi}{4}}\large\frac{\cos x-\sin x}{(\Large\frac{\pi}{4}-\normalsize x)(\large\cos x+\sin x)}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

If $f(x)=\large\frac{x^2-10x+25}{x^2-7x+10}$ for $x\neq 5$ and $f$ is continuous at $x=5$ then $f(5)=$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \large\frac{\pi}{4}}\big(\large\frac{1-\tan x}{1\sqrt 2\sin x}\big)$ =

asked Dec 30, 2013 by sreemathi.v

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If $\lim\limits_{x\to k}\large\frac{x^k-5^k}{x-5}$ $=500$ then positive value of k is

asked Dec 30, 2013 by sreemathi.v

1 answer

$f(x)=\left\{\begin{array}{1 1}\large\frac{x^5-32}{x-2}&x\neq 2\\k&x=2\end{array}\right.$ is continuous at $x=2$ then $k=$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to\large\frac{-\pi}{4}}\large\frac{1+\tan x}{\cos 2x}$ =

asked Dec 30, 2013 by sreemathi.v

1 answer

If $f$ is continuous on $[0,1]$ and $f\big(\large\frac{1}{3}\big)$ $=1$ then $\lim\limits_{n\to \infty}f(\large\frac{n}{\sqrt{9n^2+1}}\big)$ =

asked Dec 30, 2013 by sreemathi.v

1 answer

If $x_1=1$ and $x_{n+1}=\large\frac{4+3x_n}{3+2x_n}\qquad$ $n \geq 1$ and if $\lim\limits_{n\to \infty}x_n=l$ then l is

asked Dec 30, 2013 by sreemathi.v

1 answer

If $\lim\limits_{x\to \infty}\big(\large\frac{x^3+1}{x^2+1}$ $-(ax+b)\big)=2$ then

asked Dec 30, 2013 by sreemathi.v

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If $\log (x+y)=2xy$ then $y'(0)=$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{h\to 0}\large\frac{f(2h+2+h^2)-f(2)}{f(h-h^2+1)-f(1)}$ given that $f'(2)=6$ and $f'(1)=4$

asked Dec 30, 2013 by sreemathi.v

1 answer

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

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{\sin(\pi\cos^2x)}{x^2}$ =

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}\large\frac{\sqrt{1-\cos 2(x-1)}}{x-1}$ =

asked Dec 30, 2013 by sreemathi.v

1 answer

The value of $\lim\limits_{x\to 0}\int\limits_{0^{x^2}}\large\frac{\cos t^2dt}{x\sin x}$ is

asked Dec 30, 2013 by sreemathi.v

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$\lim\limits_{x\to \infty}\large\frac{x^n}{e^x}$ $=0$ for

asked Dec 30, 2013 by sreemathi.v

1 answer

If $G(x)=-\sqrt{25-x^2}$ then $\lim\limits_{x\to 1}\large\frac{G(x)-G(1)}{x-1}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{\theta \to \Large\frac{\pi}{2}}(\sec\theta-\tan\theta)$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{\tan x-\sin x}{x^3}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}[x-\sqrt{x^2+x}]$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

If $f(x)=\large\frac{1-\tan x}{4x-\pi}\qquad$ $x\neq \large\frac{\pi}{4}$ , $x \in [0,\large\frac{\pi}{2}]$ and f(x) is continuous in $[0,\large\frac{\pi}{2}]$ then $f(\large\frac{\pi}{4})$

asked Dec 24, 2013 by sreemathi.v

1 answer

If $f(5)=15,f'(5)=3$ then $\lim\limits_{x\to 5}\large\frac{xf(5)-5f'(x)}{x-5}$ is

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to a}\large\frac{\log(x-a)}{\log(e^x-e^a)}$ is

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to\Large\frac{\pi}{6}}\large\frac{3\sin x-\sqrt 3\cos x}{6x-\pi}$ equals

asked Dec 24, 2013 by sreemathi.v

1 answer

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

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}\big(\large\frac{x+a}{x+b}\big)^{x+b}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

If $f(x)=\left\{\begin{array}{1 1}\large\frac{\sin[x]}{[x]}&[x]\neq 0\\0&[x]=0\end{array}\right.$ then

asked Dec 24, 2013 by sreemathi.v

1 answer

$f(x)=\large\frac{\log(1+ax)-\log(1-bx)}{x}$ 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 Dec 24, 2013 by sreemathi.v

1 answer

If $\lim\limits_{x\to 0}\large\frac{\sin px}{\tan 3x}$ $=4$ then value of p is

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}\large\frac{(2x-3)(3x-4)}{(4x-5)(5x-6)}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

The number of points at which the function $f(x)=\large\frac{1}{\log|x|}$ is discontinuous is

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \large\frac{\pi}{4}}\large\frac{\sqrt 2\cos x-1}{\cot x-1}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

The value of $f(0)$ so that the function $f(x)=\large\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x}$ is continuous at each point on its domain is

asked Dec 24, 2013 by sreemathi.v

1 answer

If the function $f(x)=\left\{\begin{array}{1 1}\large\frac{x^2-(A+2)x+A}{x-2}&x\neq 2\\2&x=2\end{array}\right.$ is continuous at $x=2$ then

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{(1-\cos 2x)\sin 5x}{x^2\sin 3x}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{\sin x-x}{x^3}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

Let $f:R\rightarrow R$ be a differentiable function having $f(2)=6,f'(2)=\large\frac{1}{48}$ .Then $\lim\limits_{x\to 2}\int_6^{f(x)}\large\frac{4t^3dt}{x-2}$ equals

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \large\frac{\pi}{4}}\large\frac{\sqrt 2\cos x-1}{\cos x-1}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}(1+ax)^{\Large\frac{b}{x}}$ =

asked Dec 24, 2013 by sreemathi.v

1 answer

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