1. ## integral limit

Calculate :
$\displaystyle \lim_{n\rightarrow \infty} \displaystyle\int^ \frac{\pi}{2}_0 \frac{\sqrt{3} }{ctg^nx+tg^nx} \,dx$

2. Originally Posted by petter
Calculate :
$\displaystyle \lim_{n\rightarrow \infty} \displaystyle\int^ \frac{\pi}{2}_0 \frac{\sqrt{3} }{ctg^nx+tg^nx} \,dx$
Consider that $\displaystyle \left|\frac{1}{\cot^n(x)+\tan^n(x)}\right|\leqslan t{1}~\forall{x}\in\left(0,\frac{\pi}{2}\right)$, and since $\displaystyle \int_0^{\frac{\pi}{2}}dx$ converges, we can state that $\displaystyle \int_0^{\frac{\pi}{2}}\frac{\sqrt{3}}{\cot^n(x)+\t an^n(x)}dx$ converges uniformly. So $\displaystyle \lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\frac{\sqrt {3}}{\cot^n(x)+\tan^n(x)}=\int_0^{\frac{\pi}{2}}\l im_{n\to\infty}\frac{\sqrt{3}}{\cot^n(x)+\tan^n(x) }dx$. Can you go from there?

EDIT: Ill finish it. $\displaystyle \forall{x}\in\left(0,\frac{\pi}{4}\right)~\cot(x)> 1$

And $\displaystyle \forall{x}\in\left(\frac{\pi}{4},\frac{\pi}{2}\rig ht)~\tan(x)>1$

So rewrite this integral as $\displaystyle \int_0^{\frac{\pi}{4}}\lim_{n\to\infty}\frac{\sqrt {3}}{\cot^n(x)+\tan^n(x)}dx+\int_{\frac{\pi}{4}}^{ \frac{\pi}{2}}\lim_{n\to\infty}\frac{\sqrt{3}}{\co t^n(x)+\tan^n(x)}dx$

Now since on those intervals one of the functions is greater than one we have that both limits go to zero.

Thus $\displaystyle \lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\frac{\sqrt {3}}{\cot^n(x)+\tan^n(x)}dx=0$