What is limit of: $\displaystyle\lim_{x\to 0}$$\tan x - \sin x\over x$ I want to search limit of this trigonometric function:
$$\displaystyle\lim_{x\to 0}\frac{\tan x - \sin x}{x^n}$$
Note: $n \geq 1$

By Taylor expansion
$$L=\lim_{x\to0}\frac{x^3/2+O(x^5)}{x^3} \cdot x^{(3
-n)}=\lim_{x\to0}\frac{x^{(3
-n)}}{2} $$
So, for $n=1,2 \rightarrow L=0$; $n=3 \rightarrow L=1/2$, $n$-even $\ge 4$ $\rightarrow$ L doesn't exist,$n$-odd $\ge 5$ $\rightarrow L=\infty$ 
