<s>[INST] Show $\lim\limits_{n\to\infty} \dfrac{n^2+3^{2n}}{(n^3+3^n)^2} = 1$ Could you help me show that
$$\lim\limits_{n\to\infty} \dfrac{n^2+3^{2n}}{(n^3+3^n)^2} = 1$$
 [/INST] Divide numerator and denominator by $3^{2n}$, yielding:
$$\frac{\frac{n^2}{3^{2n}}+1}{\left(\frac{n^3}{3^n}+1\right)^2}$$
Prove the numerator and denominator both approach $1$ as $n\to\infty$.
This amounts to showing $\frac{n^2}{3^{2n}}\to 0$ and $\frac{n^3}{3^n}\to 0$ as $n\to\infty$.
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