Limiting question:$\displaystyle \lim_{x\to\ 0} \frac{a^{\tan\ x} - a^{\sin\ x}}{\tan\ x -\sin\ x}$ How do I find the value of $$\lim_{x\to\ 0} \frac{a^{\tan\ x} - a^{\sin\ x}}{\tan\ x - \sin\ x}$$
in easy way.

Let $u=\tan(x)-\sin(x)$. Note that as $x\to 0$, $u\to 0$.  Then, we can write
$$\begin{align}
\lim_{x\to 0}\frac{a^{\tan(x)}-a^{\sin(x)}}{\tan(x)-\sin(x)}&=\left(\lim_{x\to 0}\frac{a^{\tan(x)-\sin(x)}-1}{\tan(x)-\sin(x)}\right)\,\left(\lim_{x\to 0}a^{\sin(x)}\right)\\\\
&=\left(\lim_{u\to 0}\frac{a^u-1}{u}\right)\,\left(\lim_{x\to 0}a^{\sin(x)}\right)\\\\
&=\log(a)
\end{align}$$
since 
$$\lim_{u\to 0}\frac{a^u-1}{u}=\left.\frac{da^u}{du}\right|_{u=0}=\log(a)$$
