<s>[INST] Find and prove the limit $\lim_{n \to \infty}\frac{2^{n+2}+3^{n+3}}{2^n+3^n}$ I'm trying to find and prove the limit $$\lim_{n \to \infty}\frac{2^{n+2}+3^{n+3}}{2^n+3^n}$$
which according to WolframAlpha happens to be $27$.
The furthest I could get is
$$\frac{2^{n+2}+3^{n+3}}{2^n+3^n} = 4\left(1 + \frac{3^{n}}{2^n+3^n}\right)$$
 [/INST] $$2^{n+2}+3^{n+3}\sim_\infty 3^{n+3}, \qquad 2^n+3^n\sim_\infty 3^n,$$
hence
$$\frac{2^{n+2}+3^{n+3}}{2^n+3^n}\sim_\infty\frac{3^{n+3}}{3^n}=3^3.$$
</s>