Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 388: 115


$$\ln 2$$

Work Step by Step

Since \begin{align*} \lim _{t \rightarrow \infty} \frac{\ln (t+2)}{\log _{2} t}&=\lim _{t \rightarrow \infty} \frac{\ln (t+2)}{\ln t}\ln 2\\ &=\frac{\infty}{\infty} \end{align*} then by using L’Hôpital’s Rule \begin{align*} \lim _{t \rightarrow \infty} \frac{\ln (t+2)}{\log _{2} t}&=\lim _{t \rightarrow \infty} \frac{\ln (t+2)}{\ln t}\ln 2\\ &=\ln 2\lim _{t \rightarrow \infty} \frac{\frac{1}{t+2}}{\frac{1}{ t}}\\ &= \ln 2\lim _{t \rightarrow \infty} \frac{t}{t+2} \\ &=\ln 2 \end{align*}
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