University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Additional and Advanced Exercises - Page 112: 2

Answer

a) As $x$ approaches $\infty$, the value of $f(x)$ stays the same at about $0.367879$. b) The graph is a straight line from $x=0\to\infty$, and the $y$-value stays constantly at about $0.367879$.
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Work Step by Step

a) Using the calculator to calculate $(\frac{1}{x})^{1/\ln x}$ for $x=10, 100, 1000$ and so on: - For $x=10$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ - For $x=100$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ - For $x=1000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ - For $x=10000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ - For $x=100000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ - For $x=1000000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$ We can easily see the pattern here is that as $x$ gets larger and approaches $\infty$, the value of $(\frac{1}{x})^{1/\ln x}$ stays the same at approximately $0.367879$. b) The graph of the function $f(x)=(\frac{1}{x})^{1/\ln x}$ is included below. The graph is a straight line from $x=0\to\infty$, and the $y$-value stays constantly at about $0.367879$.
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