calculus - Show that $lim_{x to infty} frac{a^x}{x}=infty$ as $xrightarrow$ $infty$ for $a>1$

Using Cauchy and Heine (Taylor expansion, L'Hospital rule, etc... is not allowed), prove that $\lim_{x \to \infty} \frac{a^x}{x}=\infty$ as $x$ approaches $\infty$ for $a>1$.

Iv'e seen this - Proving a limit by Cauchy definition, but didn't understand how I can evaluate.

Please help, thank you!

Answer

Let $b=a-1>0$. For each natural $n>3$, if $n\leq x

$$\begin{align} \frac{a^x}x&>\frac{a^n}{n+1}\\ &=\frac{(1+b)^n}{n+1}\\ &\geq\frac{1+bn+\frac{n^2-n}2b^2}{2n-2}\\ &>\frac{b^2}4n \end{align}$$

Note: The condition $n>3$ has the only reason of that we can "transform" the denominator $n+1$ into $2n-2$ in order to simplify the factor $(n-1)$.