Which increases more quickly in the expression $\frac{n^5+\cos(n)}{2-6^n}$?

Question

I've read other answers regarding sequence limits, but it's not clear to me whether a finite value to the $n$th power, or $n$ to some finite power, would increase faster as $n$ approaches infinity.

Answer 1

Divide the numerator and denominator by the leading term in the denominator ($6^n$): $$\frac{6^{-n}n^5+6^{-n}\cos{n}}{2^{1-n}\times3^{-n}-1}$$

The limits of individual terms can be calculated:

$$\lim_{n\rightarrow\infty}6^{-n}n^5=0 \\ \lim_{n\rightarrow\infty}6^{-n}\cos{n}=0 \\ \lim_{n\rightarrow\infty}2^{1-n}=0 \\ \lim_{n\rightarrow\infty}3^{-n}=0$$

Then we have $$\frac{0+0}{0\times0-1} = \frac{0}{1} = 0$$