Mathematical analysis - problem with calculating limit

Question

I need to calculate one limit and I have big problems. I would be very thankful if someone could help me...

This is the limit I need to find: $$ \lim_{n\rightarrow\infty}\frac{2^n + n\sin{n}}{\log_2{n} + e^n} $$

Any hint/explanation would be very helpful!

Answer 1

hint: $0 < f(n) < 2\cdot \dfrac{2^n}{e^n}= 2\cdot \left(\dfrac{2}{e}\right)^n, n \ge N_0$

Answer 2

$$0\le\lim_{n\to\infty}\frac{2^n +n\sin n}{\log_2n+e^n} \le \lim_{n\to\infty}\frac{2^n +n}{e^n}=\lim_{n\to\infty}\left(\frac{2}{e}\right)^n+\lim_{n\to\infty}\frac{n}{e^n}=0+0=0.$$