The value of $\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos (1 - \cos x)}}{{x\tan x - {x^2}}}$ is:

If $\lim _{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B$,then the values of $A$ and $B$ respectively are

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 1} \frac{4x+3}{x-2}$

If $a, b$ and $c$ are three distinct real numbers and $\lim _{x \rightarrow \infty} \frac{(b-c) x^2+(c-a) x+(a-b)}{(a-b) x^2+(b-c) x+(c-a)}=\frac{1}{2}$,then $a+2 c=$

If $\mathop {\lim }\limits_{n \to \infty } \frac{1 - 10^n}{1 + 10^{n+1}} = \frac{-\alpha}{10}$,then find the value of $\alpha$.

The right hand and left hand limit of the function $f(x)$ are respectively:
$f(x)=\begin{cases} \frac{e^{1 / x}-1}{e^{1 / x}+1}, & \text{if } x \neq 0 \\ 0, & \text{if } x=0 \end{cases}$