Without using L'Hopital's rule, create the end value:

Question

Without using L'Hopital's rule, create the end value: $$\lim_{x\to 0}\frac{x-\sin x}{x-\tan x}$$

Answer 1

$$ \frac{x-\sin\left(x\right)}{x-\tan\left(x\right)}\underset{0}{=}\frac{x-x+x^3/6+o\left(x^3\right)}{x-x-x^3/3+o\left(x^3\right)}\underset{x\rightarrow 0}{\rightarrow}-\frac{3}{6}=-\frac{1}{2} $$