Example 4. Find $\lim _{x \rightarrow \infty} \frac{a^{x}}{x^{\alpha}}$ for $a>1, \alpha>0$.
Solution. When calculating this limit, we can assume $x>1$ (since $x \rightarrow \infty$) and if $n$ is the nearest integer greater than $\alpha$, then

$$
\frac{a^{x}}{x^{\alpha}}>\frac{a^{x}}{x^{n}}(n>0)
$$

Therefore,

$$
\lim _{x \rightarrow \infty} \frac{a^{x}}{x^{\alpha}} \geq \lim _{x \rightarrow \infty} \frac{a^{x}}{x^{n}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-102.jpg?height=32&width=310&top_left_y=197&top_left_x=46)

Since $n>0$, the limit of the right-hand side of the last inequality can be found using L'Hôpital's rule.

Applying this rule $n$ times, we get

$$
\lim _{x \rightarrow \infty} \frac{a^{x}}{x^{n}}=\lim _{x \rightarrow \infty} \frac{a^{x}[\ln a]^{n}}{n(n-1)(n-2) \ldots 1}=\infty
$$

Thus, if $a>1$ and $\alpha>0$, then

$$
\lim _{x \rightarrow \infty} \frac{a^{x}}{x^{\alpha}}=\infty
$$

and, consequently,

$$
\lim _{x \rightarrow \infty} \frac{x^{\alpha}}{a^{x}}=0
$$

Remark 1. The exponential function $a^{x}$ for $a>1$ grows faster than any power of $x$.

Remark 2. If $P(x)$ is any polynomial

$$
P(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}
$$

then

$$
\lim _{x \rightarrow \infty} \frac{P(x)}{a^{x}}=0
$$

and, consequently,

$$
\lim _{x \rightarrow \infty} \frac{a^{x}}{|P(x)|}=\infty
$$