Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Additional and Advanced Exercises - Page 102: 1

Answer

a. $1$, see table. b. $1$, see graph.

Work Step by Step

a. See table for the calculations; we can see that: as $x\to0$, $y\to1$. b. See graph; it can be seen that the function approaches the y-axis to a value of $y=1$ Extra: prove analytically. Given $y=x^x$, we have $x=\log_xy=\ln(y)/\ln(x)$, thus $\ln(y)=x\cdot \ln(x)=\ln(x)/(1/x)$ and using L’Hopital’s Rule, we have $\lim_{x\to0} \ln(y) =\lim_{x\to0} \ln x/(1/x)=\lim_{x\to0}(1/x)/(-1/x^2)= \lim_{x\to0}(-x)=0$ which gives $y=1$
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