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: 112



Work Step by Step

Since we have $$\lim _{t \rightarrow \infty} \frac{\ln(e^t+1)}{t}=\frac{\infty}{\infty}.$$ We can apply L’Hôpital’s Rule as follows $$\lim _{t \rightarrow \infty} \frac{\ln(e^t+1)}{t}=\lim _{t \rightarrow \infty} \frac{(e^t/(e^t+1))}{1}=\lim _{t \rightarrow \infty} \frac{e^t}{e^t+1}=\frac{\infty}{\infty}.$$ Again, we can apply L’Hôpital’s Rule as follows $$ \lim _{t \rightarrow \infty} \frac{e^t}{e^t+1}=\lim _{t \rightarrow \infty} \frac{e^t}{e^t}=1.$$ Hence, $$\lim _{t \rightarrow \infty} \frac{\ln(e^t+1)}{t}=1.$$
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