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

If $\mathop {\lim }\limits_{x \to a} \frac{{{a^x} - {x^a}}}{{{x^x} - {a^a}}} = - 1$,then

$\lim _{x \rightarrow \pi / 6} \left[ \frac{3 \sin x - \sqrt{3} \cos x}{6x - \pi} \right]$ is equal to:

If $\lim _{x}$ ${\rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}$,where $\operatorname{gcd}(m, n)=1$,then $8 m+12 n$ is equal to.

Let $l = \mathop {Lim}\limits_{x \to {0^ + }} x^m (\ln x)^n$ where $m, n \in N$,then:

If $a > 0$ and $\lim _{x \rightarrow a} \frac{a^x - x^a}{x^x - a^a} = -1$,then $a$ is equal to