Which of these function grows faster?

Question

I have the functions

$$\frac{n^3}{100000}$$

and

$$\frac{n^3}{\log_2(n)}+100n+5000n^2$$

I can't figure out which one grows faster. Can anyone help me?

Answer 1

Hint:

For all $\alpha,\beta >0$, one has

$\log_2^\beta n=o\left(n^\alpha\right)$, so there results the asymptotic equivalence $$\frac{n^3}{\log_2(n)}+100n+5000n^2\sim_\infty \frac{n^3}{\log_2(n)}.$$ can you conclude from there?

Answer 2

HINT

Divide everything by $n^3$. The top is a constant, the bottom has 3 terms, first one decays at $\log_2(n)$, the second decays quadratically and the third -- linearly. Either way all 3 decay...