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I know that it can be also evaluate using Taylor expansion, but I am intentionally want to solve it using L'Hôpital's rule:

$$ \lim\limits_{x\to 0} \frac{\sin x}{x}^{\frac{1}{1-\cos x}} = \lim\limits_{x\to 0}\exp\left( \frac{\ln(\frac{\sin x}{x})}{1-\cos x} \right)$$

Now, from continuity and L'hopital Rule: $$\lim\limits_{x\to 0} \frac{\ln(\frac{\sin x}{x})}{1-\cos x} = \lim\limits_{x\to 0} \frac{\frac{x}{\sin x}\cdot\frac{x\cos x - \sin x}{x^2}}{\sin x} = \lim\limits_{x\to 0}\frac{\frac{x\cos x - \sin x}{x\sin x}}{\sin x}$$

This is where I got stuck.
If I'm not mistaken the limit is $-\frac{1}{3}$ so the orginial one is $e^{-\frac{1}{3}}$

What should I do different (Or what's is wrong with my calculation?)
Thanks

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Have you tried applying the rule again? –  Najib Idrissi Aug 8 at 15:40

3 Answers 3

up vote 5 down vote accepted

After rearranging the quotient a bit we can apply L'hopitals rules $2$ more times to get the answer:

$$\lim_{x\to0}\frac{x\cos(x)-\sin(x)}{x\sin^{2}(x)}\underbrace{=}_{\text{l'hopital}}\lim_{x\to0}\frac{-x\sin(x)}{\sin^{2}(x)+2x\sin(x)\cos(x)}=\lim_{x\to0}\frac{-x}{\sin(x)+2x\cos(x)}$$

$$\underbrace{=}_{\text{l'hopital}}\lim_{x\to0}\frac{-1}{3\cos(x)-2x\sin(x)}=\frac{-1}{3}$$

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Thanks, I should have just continue and not give up :) –  Elimination Aug 8 at 15:56
    
You're welcome. –  user71352 Aug 8 at 16:01

You can continue it as the following : $$\begin{align}\lim_{x\to 0}\frac{x\cos x-\sin x}{x\sin^2 x}&=\lim_{x\to 0}\frac{(x\cos x-\sin x)'}{(x\sin^2 x)'}\\&=\lim_{x\to 0}\frac{\color{red}{\cos x}-x\sin x\color{red}{-\cos x}}{\sin^2 x+2x\sin x\cos x}\\&=\lim_{x\to 0}\frac{-x}{\sin x+2x\cos x}\\&=\lim_{x\to 0}\frac{-1}{(\sin x/x)+2\cos x}\\&=\frac{-1}{1+2\cdot 1}.\end{align}$$

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Thank you @mathlove –  Elimination Aug 8 at 15:58
    
@Elimination: You are welcome. –  mathlove Aug 8 at 16:05

Alternative approach using Taylor series:

Using $\frac{\sin x}{x} = 1 - \frac{x^2}{6} + o(x^2)$ we get $\ln(\frac{\sin x}{x}) \sim_0 \ln(1-\frac{x^2}{6})$. This expression has the Taylor expansion $-\frac{x^2}{6}+o(x^2)$.

We can approximate the denominator by $1-\cos x=\frac{x^2}{2} + o(x^2)$.

Then we have: $\frac{\ln(\frac{\sin x}{x})}{1-\cos x} \sim_0 -\frac{x^2}{6} \frac{2}{x^2} = -\frac{1}{3}$.

This doesn't exactly answer your question, but hopefully it provides a little bit of insight. :)

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