Question

Solve: $\underset{x\to 2}{lim}\frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x},a>0$

Answer 1

$\underset{x\to 2}{lim}\frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x}$

$=\left(\underset{x\to 2}{lim}{a}^{\tan x}-\underset{x\to 2}{lim}{a}^{\sin x}\right)\times \frac{1}{\underset{x\to 2}{lim}\tan x-\underset{x\to 2}{lim}\sin x}$

$=\frac{a^{\tan 2}-a^{\sin 2}}{\tan 2-\sin 2}$

$\therefore \underset{x\to 2}{lim}\frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x}=\frac{a^{\tan 2}-a^{\sin 2}}{\tan 2-\sin 2}$.