<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] $$\frac{2^{n+2}+3^{n+3}}{2^n+3^n}=\frac{(4)2^{n}+(27)3^{n}}{2^n+3^n}=\frac{(4)2^{n}/3^n+(27)1}{2^n/3^n+1}$$
so the limit is $27$
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