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It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.

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$$\lim_{n\to\infty} \frac{3^n}{2^n+3^n}=\lim_{n\to \infty}\frac{1}{(\frac{2}{3})^n+1} =1.$$Since $\frac{2}{3}<1$ , so $(\frac{2}{3})^n\to 0$ as $n\to \infty$.

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