TITLE: Limit of $\arctan(x)/x$ as $x$ approaches $0$?
QUESTION [6 upvotes]: Quick question:
I came across the following limit: $$\lim_{x\rightarrow 0^{+}}\frac{\arctan(x)}{x}=1.$$ 
It seems like the well-known limit: 
$$\lim_{x\rightarrow 0}\frac{\sin x}{x}=1.$$ 
Can anyone show me how to prove it?

REPLY [7 votes]: Since $\lim\limits_{x\to0}\arctan(x)=0$, letting $x=\tan(\theta)$ yields
$$
\lim_{x\to0}\frac{\arctan(x)}{x}=\lim_{\theta \to0}\frac{\theta}{\tan(\theta)}\tag{1}
$$
and $(1)$ is shown to be $\frac11$ in equation $(5)$ of this answer.