Find $\lim_{x\to 0} \frac{\arcsin x}{x}$

Question

Do I have to use L'hoptial's rule? How can this be used in this instance?

Answer 1

Let $x = \sin y$. Now, $x \to 0 \Rightarrow y \to 0$, giving us the following limit:

$$ \lim\limits_{y \to 0} \frac{\arcsin (\sin y)}{\sin y} = \lim\limits_{y \to 0} \frac{y}{\sin y} = \frac{1}{\lim\limits_{y \to 0} \frac{\sin y}{y}} = 1 $$

Answer 2

we have $$\lim_{x\to0}\frac{\arcsin x}{x}$$ making $u=\arcsin x,x=\sin u,x\mapsto0= u\mapsto0$ $$\lim_{x\to0}\frac{\arcsin x}{x}=\lim_{u\to 0}\frac{u}{\sin u}=\lim_{u\to 0}\frac{1}{\frac{\sin u}{u}}=\frac{1}{\displaystyle\lim_{u\to0}\frac{\sin u}{u}}=1$$ you can use l'hospital since its a limit of type 0/0 wich gives $$\lim_{x\to 0}\frac{\arcsin x}{x}= \lim_{x\to 0}\frac{1}{\sqrt{1-x^2}}=1$$