Limit proof

• Jul 18th 2011, 05:40 AM
Siron
Limit proof
Hi, I've to proof that:

$\displaystyle \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=1$

This is what I found:

$\displaystyle \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=\lim_{x \to 0} \left(\int_{x}^{\frac{\pi}{2}} \frac{du}{\sin^2(u)} - \int_{x}^{\frac{\pi}{2}}\frac{\cos(u)du}{\sin^2(u) }\right)$
$\displaystyle =\lim_{x \to 0} \left( \left[-\cot(u)\right]_{x}^{\frac{\pi}{2}}+\left[\frac{1}{\sin(u)}\right]_{x}^{\frac{\pi}{2}}\right)$
$\displaystyle =\lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)$

The problem is here, because $\displaystyle \lim_{x \to 0} \cot(x)=+\infty$ and $\displaystyle \lim_{x \to 0} \frac{1}{\sin(x)}=\infty}$ I don't now how to go further.

• Jul 18th 2011, 05:53 AM
Sudharaka
Re: Limit proof
Quote:

Originally Posted by Siron
Hi, I've to proof that:

$\displaystyle \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=1$

This is what I found:

$\displaystyle \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=\lim_{x \to 0} \left(\int_{x}^{\frac{\pi}{2}} \frac{du}{\sin^2(u)} - \int_{x}^{\frac{\pi}{2}}\frac{\cos(u)du}{\sin^2(u) }\right)$
$\displaystyle =\lim_{x \to 0} \left( \left[-\cot(u)\right]_{x}^{\frac{\pi}{2}}+\left[\frac{1}{\sin(u)}\right]_{x}^{\frac{\pi}{2}}\right)$
$\displaystyle =\lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)$

The problem is here, because $\displaystyle \lim_{x \to 0} \cot(x)=+\infty$ and $\displaystyle \lim_{x \to 0} \frac{1}{\sin(x)}=\infty}$ I don't now how to go further.

$\displaystyle \lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)=\lim_{x \to 0} \left(\frac{\cos x+\sin x-1}{\sin x}\right)$