<s>[INST] How to calculate $\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$ I came across this strange limit whilst showing convergence of a series:
$$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$$
How can I calculate this limit?
 [/INST] Use the sandwich/squeeze theorem. Since
$$\frac{3^n}{2\cdot 4^n}<\frac{2^n + 3^n}{3^n + 4^n} < \frac{2\cdot 3^n}{4^n}$$
for all $n$,
$$\frac{1}{2^{1/n}} \cdot \frac{3}{4} < \sqrt[n]{\frac{2^n + 3^n}{3^n + 4^n}} < 2^{1/n}\cdot \frac{3}{4}$$
for all $n$. Since $2^{1/n} \to 1$ as $n\to \infty$, by the squeeze theorem, your limit is $3/4$.
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