<s>[INST] find limit of a sequence I have a sequence:
$$ X_n\:=\:\cos\left(\left(\frac{3^n+\pi^n}{3^n+\left(\pi-\frac{1}{4} \right)^n} \right)^{1/n}\right) $$
I have to find the limit when $n \to \infty $ where $n \in \mathbb{N}$.
Which is the best way to find the answer ? Can I reduce or use the Squeeze theorem in that case ?
 [/INST] $$a_n=\sqrt[n]{\frac{3^n+ \pi^n}{3^n+ (\pi -1/4)^n}}\leq \sqrt[n]{\frac{2 \pi^n}{3^n}}=\sqrt[n]{2} \frac{\pi}{3} \to \frac{\pi}{3}$$
Also $$a_n=\sqrt[n]{\frac{3^n+ \pi^n}{3^n+ (\pi -1/4)^n}} \geq \sqrt[n]{ \frac{\pi^n}{3^n+3^n}}=\frac{1}{\sqrt[n]{2}} \frac{\pi}{3} \to \frac{\pi}{3}$$
So from squeeze theorem $a_n \to \frac{\pi}{3}$.
Now using the continuity of $\cos{x}$ we have that $$\cos{a_n} \to \cos{\frac{\pi}{3}}=\frac{1}{2}$$
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