Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} \frac{\cos x}{\pi - x}$

If $\alpha$ is the interior angle of a regular octagon,then $\lim_{\theta \to \alpha^+} \frac{\tan \theta - 1}{[\sin \theta + \cos \theta]}$ is equal to (Note: $[k]$ denotes the greatest integer less than or equal to $k$).

If $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-x+1}-a x\right)=b$,then the ordered pair $(a, b)$ is:

Let $A = \lim_{x \rightarrow 0^{+}} \left(1 + \tan^2 \sqrt{x}\right)^{\frac{1}{2x}}$,then $\log_{e} A = $

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

Evaluate the limit: $\lim _{x}$ ${\rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3$