Limits

Given a real valued function ' $f$ ' such that $f(x)=\left\{\begin{array}{cc}\frac{\tan ^2\{x\}}{x^2-[x]^2} & \text { for } x\gt0 \\ 1 & \text { for } x=0 \\ \sqrt{\{x\} \cot \{x\}} & \text { for } x \lt…

The value of $\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4}$ is

If $\lim _{x \rightarrow} x \sin \left(\frac{1}{x}\right)=A$ and $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=B$, then which one of the following is correct?

If $\text{lim} n→∞(an-\frac{1+{n}^{2}}{1+n})=b$, where $a$ is finite number, then

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

Value of $\lim _{x \rightarrow 0} \frac{a^{\sin x}-1}{\sin x}$ is

$\lim _{x \rightarrow \pi / 2} \frac{\left(1-\tan \left(\frac{x}{2}\right)\right)(1-\sin x)}{\left(1+\tan \left(\frac{x}{2}\right)\right)(\pi-2 x)^{3}}=?$

$\text{lim} n→∞\sin[π\sqrt{{n}^{2}+1}]$ is equal to

Two tangents \(P Q\) and \(P R\) drawn to the circle \(x^2+y^2-2 x-4 y-20=0\) from point \(P(16,7)\). Ifthe centre of the circle is \(C\) then the area of quadrilateral \(\mathrm{PQCR}\) is

Evaluate \(\lim _{x \rightarrow \infty} \frac{\sqrt{1+\sin 3 x}-1}{\ln (1+\tan 2 x)}\);

$\lim _{x \rightarrow 0} \frac{|\sin x|}{x}$ is equal to

$\lim _{x \rightarrow 0}(\operatorname{cosec} x)^{1 / \log x}$ is equal to :