Mathematics - Limits Question with Solution | TestHub

We have ${\lim }_{x\to 0}\frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x}={\lim }_{x\to 0}{a}^{\sin x}\left(\frac{a^{\tan x-\sin x}-1}{\tan x-\sin x}\right)$ $\begin{array}{rlr}& =\underset{x\to 0}{\lim }\left(a^{\sin x}\right)\times \underset{t\to 0}{\lim }\left(\frac{a^t-1}{t}\right)(\text{ where }t=\tan x-\sin x)& =a^0\times {\log }_{e}a={\log }_{e}a\end{array}$