If $f(0) = 2,$ then $\mathop {\lim }\limits_{x \to 0} \frac{{\int\limits_0^x {\left( {tf(x) + xf(t)} \right)dt} }}{{{x^2}}}$ is equal to -

$\lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{3^{x}-1}$ is equal to

$\lim _{x \rightarrow 0} \left( \frac{3^{x}-1}{x} \right)$ is equal to

The value of the limit $\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - {e^{ - x}} - 2x}}{{x - \sin x}}$ is

If $\alpha = \lim_{x \rightarrow \pi/4} \frac{\tan^{3} x - \tan x}{\cos(x + \pi/4)}$ and $\beta = \lim_{x \rightarrow 0} (\cos x)^{\cot x}$ are the roots of the equation $ax^{2} + bx - 4 = 0$,then the ordered pair $(a, b)$ is:

The value of $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\int_{\frac{\pi }{2}}^x t \,dt}}{{\sin (2x - \pi )}}$ is