Question

lim as x goes to inf (cosx)^(1/x^2)

Answer 1

Recall that cosine is bounded by -1 and 1.
\[\large\rm -1\le \cos x\le 1\]If we raise each side to the 1/x^2 power, we get,\[\large\rm (-1)^{1/x^2}\le (\cos x)^{1/x^2}\le 1^{1/x^2}\]

\(\large\rm (-1)^{1/x^2}\to1\)

\(\large\rm 1^{1/x^2}\to1\)

Squeeze Theorem does the rest, ya? :)

Answer 2

no

Answer 3

the limit does not exist

Answer 4

\[\lim_{x\to\infty}(\cos x)^{1/x^2}=\exp\left(\lim_{x\to\infty}\ln\frac{\cos x}{x^2}\right)=\exp\left(\color{red}{\lim_{x\to\infty}\ln\cos x}-\lim_{x\to\infty}x^2\right)\]