<s>[INST] Limit of this sequence $ \lim_{n\to \infty}\sqrt[n]{\frac{2^n + 3^n + 4^n}{5^n + 6^n}}$? How to find the limit of this sequence. I tried using the squeeze theorem but no result yet.
$ \lim_{n\to \infty}\sqrt[n]{\frac{2^n + 3^n + 4^n}{5^n + 6^n}}$
 [/INST] It is easy to see that $$\frac{1}{2}(\frac{4^n}{6^n})<\frac{2^n+3^n+4^n}{5^n+6^n}<3(\frac{4^n}{6^n}).$$ Therefore,
$$\frac{4}{6}\sqrt[n]{\frac{1}{2}}<(\frac{2^n+3^n+4^n}{5^n+6^n})^\frac{1}{n}<\sqrt[n]{3}\frac{4}{6}.$$ Since $\sqrt[n]{c}\to 1$ as $n\to\infty$, we have that $$\lim_{n\to\infty}(\frac{2^n+3^n+4^n}{5^n+6^n})^\frac{1}{n}=\frac{4}{6}=\frac{2}{3}.$$
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