Compute the following limit:
$ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$
I'm looking for an easy approach if possible.
Compute the following limit:
$ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$
I'm looking for an easy approach if possible.
$\arctan \left(\frac{n}{x^{2}-nx+1}\right) = \arctan(x) + \arctan(n-x)$
$ I= \int_{0}^{n} {\frac {\arctan(x)}{ \arctan(x)+\arctan(n-x)}dx} =\int_{0}^{n} {\frac {\arctan(n-x)}{ \arctan(x)+\arctan(n-x)}dx} $
$I = \frac{1}{2}\cdot\int_{0}^{n} 1dx = \frac{n}2$
$\lim_{n \to \infty }\frac1n \int_{0}^{n} \frac {\arctan(x)}{ \arctan \left(\frac{n}{x^{2}-nx+1}\right)}dx = \frac12 $