Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.3 - Calculating Limits Using the Limit Laws - 2.3 Exercises: 39

Answer

Apply the squeeze theorem, we can prove that $\lim\limits_{x\to0}x^4\cos\frac{2}{x}=0$

Work Step by Step

We know that $-1\leq\cos\frac{2}{x}\leq1$ Multiply by $x^4$ throughout, $-x^4\leq x^4\cos\frac{2}{x}\leq x^4$ (the inequality direction remains, because $x^4\geq0$ for $\forall x\in R$) Since $\lim\limits_{x\to0}x^4=0^4=0$ and $\lim\limits_{x\to0}-x^4=-0^4=0$ Therefore, applying the squeeze theorem, we have $\lim\limits_{x\to0}x^4\cos\frac{2}{x}=0$
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