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?

Hint:
$$
\sqrt[\large n]{\frac{2^n+3^n}{3^n+4^n}}=\frac34\ \sqrt[\large n]{\frac{1+\left(\frac23\right)^n}{1+\left(\frac34\right)^n}}
$$
