Let $f, g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x)=\{\begin{array}{cc}\frac{x}{|x|},& x\ne 0\\ 1,& x=0\end{array}$ $g(x)=\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)},& x\ne -1\\ 1,& x=-1\end{array}$ and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\le x$. Then the value of $\underset{x\to 1}{\lim }g(h(x-1))$ is :

Suppose $$f:\mathbb{R}\to (0,\infty )$$ be a differentiable function such that $$5f(x+y)=f(x)\cdot f(y),\forall x,y\in \mathbb{R}$$. If $$f(3)=320$$, then \sum_\limits{n=0}^{5} f(n) is equal to :

Let $$x=2$$ be a root of the equation $$x^2+px+q=0$$ and f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right. Then $$\underset{x\to 2p^{+}}{\lim }[f(x)]$$, where $$\left[.\right]$$ denotes greatest integer function, is

If the function f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 is continuous atx = {$\pi$ \over 2}$$,then$$9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$$ is equal to

The value of \mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + 3} - \sqrt {{n^4} + 5n + 4} }} is :

The set of all values of $$a$$ for which $$\underset{x\to a}{\lim }([x-5]-[2x+2])=0$$, where [$$\alpha$$] denotes the greatest integer less than or equal to $$\alpha$$ is equal to

\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t}}}}} \right)^{{{\sin }^2}t}} is equal to

Let f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right. Then at $$x=0$$

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x],2+x,x+2[x]\},0\le x\le 2$. Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not differentiable. Then $(m+n{)}^2+2$ is equal to :

If \lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17, then $$5a^2+b^2$$ is equal to

Let $$f$$ and $$g$$ be two functions defined by f(x)=\left\{\begin{array}{cc}x+1, & x Then(g \circ f)(x)$$ is :

Let $$f(x)=\left[x^2-x\right]+|-x+[x]|$$, where $$x\in \mathbb{R}$$ and $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then, $$f$$ is :

If $$\alpha >\beta >0$$ are the roots of the equation $$a{x}^2+bx+1=0$$, and \lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^{2}+b x+a\right)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right), \text { then } \mathrm{k} \text { is equal to } :

\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+1)\right)^{5}}\right)\right) is equal to _____________.

Let $$a_1,a_2,a_3,\dots ,a_{\mathrm{n}}$$ be $$\mathrm{n}$$ positive consecutive terms of an arithmetic progression. If $$\mathrm{d}>0$$ is its common difference, then \lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right) is

Let S = \left\{ {x \in R:0 If\mathrm{n(S)}$$denotesthenumberofelementsin$$\mathrm{S}$$ then :

Let (a, b) $\subset (0,2\pi )$ be the largest interval for which ${\sin }^{-1}(\sin \theta )-{\cos }^{-1}(\sin \theta )>0,\theta \in (0,2\pi )$, holds. If $\alpha x^2+\beta x+{\sin }^{-1}\left(x^2-6x+10\right)+{\cos }^{-1}\left(x^2-6x+10\right)=0$ and $\alpha -\beta =b-a$, then $\alpha$ is equal to :

Let $$S$$ be the set of all solutions of the equation $${\cos }^{-1}(2x)-2{\cos }^{-1}\left(\sqrt{1-x^2}\right)=\pi ,x\in \left[-\frac{1}{2},\frac{1}{2}\right]$$. Then \sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right) is equal to :

If $${\sin ^{ - 1}}{$\alpha$ \over {17}} + {\cos ^{ - 1}}{4 \over 5} - {\tan ^{ - 1}}{{77} \over {36}} = 0,0