How to calculate $\lim_{n \rightarrow \infty}$ $ n^{b}/a^{n}$

Question

$\lim_{n \rightarrow \infty}$ $ n^{b}/a^{n} $

I have tried to approach it using L Hopital but it is not working. Maybe using sandwich could work but i cant think of the function to enclose it

Answer 1

Assume that $b>0$ and $c>1$, then $c^{n}=(1+p)^{n}\geq\dfrac{n(n-1)}{2}p^{2}$, where $p=1-c>0$, and we have if $a>1$ and $c=a^{1/b}$ that $n^{b}/a^{n}=(n/(a^{1/b})^{n})^{b}=(n/c^{n})^{b}\leq(2/(n-1)p^{2})^{b}\rightarrow 0$ as $n\rightarrow\infty$.

Answer 2

The cases: $$\lim_\limits{n\to+\infty} \frac{n^b}{a^n}=\begin{cases} 0,\ \ \ \ \ \ \ a>1 \\ +\infty, \ \ a=1,b>0 \\ 1, \ \ \ \ \ \ \ a=1,b=0 \\ 0, \ \ \ \ \ \ \ a=1,b<0 \\ +\infty, \ \ 0<a<1 \\ +\infty, \ \ a=0,b\ne0 \\ \emptyset, \ \ \ \ \ \ \ a=0,b=0 \end{cases}$$