Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 3 - Determining Change: Derivatives - 3.7 Activities - Page 244: 10


$$\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{2}}=\infty$$

Work Step by Step

Given $$\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{2}}$$ using the Limit Rules and replacement, leads to the indeterminate form $$\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{2}}=\frac{\infty}{\infty}$$ Applying L'Hôpital's Rule \begin{align*} \lim _{x \rightarrow \infty} \frac{e^{x}}{x^{2}}&=\lim _{x \rightarrow \infty} \frac{e^{x}}{2x }\\ &=\frac{\infty}{\infty} \end{align*} Applying L'Hôpital's Rule again \begin{align*} \lim _{x \rightarrow \infty} \frac{e^{x}}{2x }&=\lim _{x \rightarrow \infty} \frac{e^{x}}{2 }\\ &= \infty \end{align*}
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