Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

Question

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.

Answer 1

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