Mathematics - Limits Question with Solution | TestHub

We have ${\lim }_{x\to 0}\frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x}={\lim }_{x\to 0}{a}^{\sin x}\left(\frac{a^{\tan x-\sin x}-1}{\tan x-\sin x}\right)$

$={\lim }_{x\to 0}\left(a^{\sin x}\right)\times {\lim }_{t\to 0}\left(\frac{a^t-1}{t}\right)$ (where $t=\tan x-\sin x)$

$=a^0\times {\log }_{e}a={\log }_{e}a$