real analysis - $ text{ Find } lim_{nrightarrow infty} frac{n}{a^n} $

$\text{Find }$ $$ \\ \lim_{n\rightarrow \infty} \frac{n}{a^n} \\ \text{ where a > 0 is a number.} $$

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$\text{To solve this problem we first realize that we need to use l'hopitals rule} \\ \text{because if we evaluate at infinity we get } \\ $

$$\lim_{n\rightarrow \infty} \frac{n}{a^n} = \frac{\infty}{\infty} \\ \text{using l'hopitals rule we get } \\ \lim_{n \rightarrow \infty}\frac{1}{n \cdot x^{n-1}} = 0 \\ \text{If we take lhopitals rule again we get } \\ \lim_{n \rightarrow \infty} \frac{0}{(n-1) \cdot n \cdot x^{n-2}} =0 \\ \text{Thus } \ \ \lim_{n\rightarrow \infty} \frac{n}{a^n} = 0 \\ \text{Is this the correct procedure that one uses in order to find the limit for this problem?} \\ \text{Is there any mistakes in my solution?} $$

Answer

L'Hospital's rule is only applicable when you have a limit of the form

$$\frac{\infty}{\infty}$$

or

$$\frac{0}{0}$$

Thus we need to consider $3$ cases: (a) $0, (b) $a=1$, and (c) $a>1$.

(a) For $a<1$ we have

$$\lim_{n\to\infty}\frac{n}{a^n}=\infty$$

because $\lim_{n\to\infty}a^n=0$ and $\lim_{n\to\infty}n=\infty$, so we have

$$\frac{\infty}{0}=\infty, \quad 0

(b) When $a=1$, we have

$$\lim_{n\to\infty}\frac{n}{a^n}=\lim_{n\to\infty}\frac{n}{1^n}=\lim_{n\to\infty}n=\infty, \quad a=1$$

(c) For $a>1$ we have

$$\lim_{n\to\infty}\frac{n}{a^n}=\frac{\infty}{\infty}$$

This is the only case where L'Hospital's rule is applicable. Here, write

$$\begin{align}\lim_{n\to\infty}\frac{n}{a^n}&=\lim_{n\to\infty}\frac{\text{d}}{\text{d}n}\left(\frac{n}{a^n}\right)\\

&=\lim_{n\to\infty}\frac{\text{d}}{\text{d}n}\left(\frac{n}{\exp(\ln(a^n))}\right)\\
&=\lim_{n\to\infty}\frac{\text{d}}{\text{d}n}\left(\frac{n}{\exp(n\ln(a))}\right)\\
&=\lim_{n\to\infty}\frac{\text{d}}{\text{d}n}\left(\frac{1}{\ln(a)\exp(n\ln(a))}\right)\\
&=\lim_{n\to\infty}\frac{\text{d}}{\text{d}n}\left(\frac{1}{\ln(a)a^n}\right)\\
&=\frac{1}{\infty}\\
\lim_{n\to\infty}\frac{n}{a^n}&=0, \quad a>1
\end{align}$$

I added extra steps to demonstrate how to perform the derivative of $a^n$.