Calculus (3rd Edition)

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

Chapter 2 - Limits - Chapter Review Exercises - Page 94: 19

Answer

The limit $ \lim _{t \rightarrow 9} \frac{t-6}{\sqrt{t}-3} $ does not exist. The one-sided limits are infinite.

Work Step by Step

By substitution, we get $$ \lim _{t \rightarrow 9} \frac{t-6}{\sqrt{t}-3}=\frac{9-6}{3-3}=\frac{3}{0} $$ which means that the limit does not exist. The one-sided limits can be calculated as follows $$ \lim _{t \rightarrow 9^+} \frac{t-6}{\sqrt{t}-3}=\frac{9-6}{3^+-3}=\infty $$ and $$ \lim _{t \rightarrow 9^-} \frac{t-6}{\sqrt{t}-3}=\frac{9-6}{3^--3}=-\infty $$ hence the limit $ \lim _{t \rightarrow 9} \frac{t-6}{\sqrt{t}-3} $ does not exist.
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