If $\lim _{x \rightarrow 1} \frac{\sin \left(3 x^2-4 x+1\right)-x^2+1}{2 x^3-7 x^2+a x+b}=-2$, then the value of $(a-b)$ is equal to

The value of $\lim _{x \rightarrow 1} \frac{\left(x^2-1\right) \sin ^2(\pi x)}{x^4-2 x^3+2 x-1}$ is equal to:

$\lim _{n \rightarrow \infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\ldots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$ is equal to

$\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^4}$ is equal to :

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\tan ^2 x\left(\left(2 \sin ^2 x+3 \sin x+4\right)^{\frac{1}{2}}-\left(\sin ^2 x+6 \sin x+2\right)^{\frac{1}{2}}\right)\right)$ is equal to

Let a be an integer such that $\lim _{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}$ exists, where $[t]$ is greatest integer $\leq t$. Then a is equal...

$\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to :

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

If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $x=0$, then $f(0)$ is equal to $\_\_\_\_$ συά

Given below are two statements: Statement I : $\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}$ Statement II: $\lim _{x \rightarrow 1}\left(x^{\frac{2}{1-x}}\right)=\frac{1}{e^2}$ In the...