Limit involving inverse trig functions

Question

I am trying to evaluate the following limit:

$$\lim \limits_{x' \to x} \frac{\sin^{-1} (y / \sqrt{x'^2 + y^2}) - \sin^{-1} (y / \sqrt{x^2 + y^2})}{x' - x}$$

I'm not sure how to get rid of the $x' - x$. I can't use L'Hopital's, by the way. I tried using $\sin^{-1}z = z + O(z^3)$, but this didn't work out.

Maybe you can change variables $y=r\sin\theta$, $x=r\cos\theta$, $x'=r\cos(\theta+\delta)$. — Suzu Hirose, Nov 12 '14 at 04:51
@Soke, Use http://math.stackexchange.com/questions/672575/proof-for-the-formula-of-sum-of-arcsine-functions-arcsin-x-arcsin-y — lab bhattacharjee, Nov 12 '14 at 04:53
Are you allowed or forbidden to use L'Hopital's ? When you say "I can't use L'Hopital's" do you mean that you don't know how to do it. Sorry but this is not clear to me. — Claude Leibovici, Nov 12 '14 at 07:55

score 1 · Answer 1 · answered Nov 12 '14 at 04:51

1

Note that it is the definition of derivative of $f(x) = \sin^{-1}\left(\frac{y}{\sqrt{x^2+y^2}}\right)$ at $x$. Use the Chain Rule.

answered Nov 12 '14 at 04:51

DeepSea

Yea, I know. I'm finding the derivative using the limit. Sorry if this was unclear. – MT_ Nov 12 '14 at 04:53

1 Answers1