Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.7 Limits at Infinity - Exercises - Page 83: 39



Work Step by Step

We wish to find the limit $$\lim _{t \rightarrow \infty} \tan \left(\frac{\pi 3^{t}+1}{4-3^{t+1}}\right) $$ Thus, we have: \begin{align*} \lim _{t \rightarrow \infty} \tan \left(\frac{\pi 3^{t}+1}{4-3^{t+1}}\right)&=\lim _{t \rightarrow \infty} \tan \left(\frac{\pi 3^{t}/3^{t}+1/3^{t}}{4/3^{t}-3^{t+1}/3^{t}}\right)\\ &=\lim _{t \rightarrow \infty} \tan \left(\frac{\pi+3^{-t}}{4 \cdot 3^{-t}-3}\right)\\ &=\tan \left(\frac{\pi}{-3}\right) \\ &=-\sqrt{3} \end{align*}
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