# How do I evaluate: lim_(x->oo) cos^(-1)((1+x^2)/(1+2x^2))?

Oct 13, 2017

We can play with a little algebra here to make the limit easier to evaluate:

${\lim}_{x \to \infty} {\cos}^{-} 1 \left(\frac{1 + {x}^{2}}{1 + 2 {x}^{2}}\right)$

$= {\lim}_{x \to \infty} {\cos}^{-} 1 \left(\frac{1 + {x}^{2}}{1 + 2 {x}^{2}} \cdot \frac{\frac{1}{x} ^ 2}{\frac{1}{x} ^ 2}\right)$

$= {\lim}_{x \to \infty} {\cos}^{-} 1 \left(\frac{\frac{1}{x} ^ 2 + \frac{{x}^{2}}{x} ^ 2}{\frac{1}{x} ^ 2 + \frac{2 {x}^{2}}{x} ^ 2}\right)$

$= {\lim}_{x \to \infty} {\cos}^{-} 1 \left(\frac{\frac{1}{x} ^ 2 + 1}{\frac{1}{x} ^ 2 + 2}\right)$

As $x \to \infty$, the first terms in the numerator and denominator go to zero.

Thus:

${\lim}_{x \to \infty} {\cos}^{-} 1 \left(\frac{\frac{1}{x} ^ 2 + 1}{\frac{1}{x} ^ 2 + 2}\right)$

$= {\cos}^{-} 1 \left(\frac{0 + 1}{0 + 2}\right)$

$= {\cos}^{-} 1 \left(\frac{1}{2}\right)$

$= \frac{\pi}{3}$