Question

$\underset{x\to 0}{\lim }\frac{a^{mx}-1}{b^{nx}-1},n\ne 0$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left(\frac{a^{mx}-1}{b^{nx}-1}\right) \\ =\underset{x\to 0}{\lim }\left(\left(\frac{a^{mx}-1}{mx}\right)\times \frac{mx}{\left({\displaystyle \frac{b^{nx}-1}{nx}}\right)\times nx}\right) \\ =\frac{{\log }_e\left(a\right)}{{\log }_e\left(b\right)}\times \frac{m}{n} \\ =\frac{m}{n}\frac{\log a}{\log b} \end{gathered}$$