$\mathop {\lim }\limits_{x \to 0} \frac{{\log \cos x}}{x} = $

If $\lim _{x \rightarrow 0} \frac{ae^{x}-b \cos x + ce^{-x}}{x \sin x} = 2,$ then $a + b + c$ is equal to ...........

If $f(4) = g(4) = 2$,$f'(4) = 9$,and $g'(4) = 6$,then $\mathop {\text{Limit}}\limits_{x \to 4} \frac{\sqrt{f(x)} - \sqrt{g(x)}}{\sqrt{x} - 2}$ is equal to:

If $f''(x)$ is continuous at $x = 0$ and $f''(0) = 4$,then find the value of $\lim_{x \to 0} \frac{2f(x) - 3f(2x) + f(4x)}{x^2}$.

If $\mathop {\lim }\limits_{x \to 0} \frac{{\log (3 + x) - \log (3 - x)}}{x} = k,$ then the value of $k$ is

$\mathop {\lim }\limits_{x \to \infty } \frac{{{x^n}}}{{{e^x}}} = 0$ for