Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.7 Exercises - Page 564: 3

Answer

\[0\]

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{x \to \infty } {x^5}{e^{ - x/100}} \hfill \\ {\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\ x = 1 \to f\left( 1 \right) = {\left( 1 \right)^5}{e^{ - \left( 1 \right)/100}} \approx 0.9900 \hfill \\ x = 10 \to f\left( {10} \right) = {\left( {10} \right)^5}{e^{ - \left( {10} \right)/100}} \approx 90483.7418 \hfill \\ x = {10^2} \to f\left( {{{10}^2}} \right) = {\left( {{{10}^2}} \right)^5}{e^{ - \left( {{{10}^2}} \right)/100}} \approx 3678794412 \hfill \\ x = {10^3} \to f\left( {{{10}^3}} \right) = {\left( {{{10}^3}} \right)^5}{e^{ - \left( {{{10}^3}} \right)/100}} \approx 4.539 \times {10^{10}} \hfill \\ x = {10^4} \to f\left( {{{10}^4}} \right) = {\left( {{{10}^4}} \right)^5}{e^{ - \left( {{{10}^4}} \right)/100}} \approx 3.7200 \times {10^{ - 24}} \hfill \\ x = {10^5} \to f\left( {{{10}^5}} \right) = {\left( {{{10}^5}} \right)^5}{e^{ - \left( {{{10}^5}} \right)/100}} \approx 0 \hfill \\ \boxed{\begin{array}{*{20}{c}} x&{f\left( x \right)} \\ 1&{0.9900} \\ {10}&{90483.7418} \\ {{{10}^2}}&{3678794412} \\ {{{10}^3}}&{4.539 \times {{10}^{10}}} \\ {{{10}^4}}&{3.7200 \times {{10}^{ - 24}}} \\ {{{10}^5}}&0 \end{array}} \hfill \\ {\text{Therefore,}} \hfill \\ \mathop {\lim }\limits_{x \to \infty } {x^5}{e^{ - x/100}} \approx 0 \hfill \\ {\text{Graph}} \hfill \\ \end{gathered} \]
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