Compute the limit of the given function

by RoRi

December 25, 2015

Evaluate the limit.

$\lim_{x \to +\infty} \frac{a^x}{x^b}, \qquad a > 1.$

We compute,

$\begin{align*} \lim_{x \to +\infty} \frac{a^x}{x^b} &= \lim_{x \to +\infty} \frac{e^{x \log a}}{x^b} \\[9pt] &= \lim_{x \to +\infty} \frac{1}{\frac{x^b}{e^{x \log a}}} = + \infty \end{align*}$

since $\lim_{x \to + \infty} \frac{x^b}{e^{x \log a}} = 0$ by Theorem 7.11.

Stumbling Robot

Compute the limit of the given function

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