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 - Section 2.4 - One-Sided Limits - Exercises 2.4 - Page 75: 18


(a) $ \sqrt{2}$ (b) $-\sqrt{2}$

Work Step by Step

(a) \begin{aligned} \lim _{x \rightarrow 1^{+}} \frac{\sqrt{2 x}(x-1)}{|x-1|} &=\lim _{x \rightarrow-1^{+}} \frac{\sqrt{2 x}(x-1)}{(x-1)}, \quad(|x-1 |=x-1 \text { for } x>1) \\ &=\lim _{x \rightarrow-1^{+}} \sqrt{2 x}\\ &=\sqrt{2} \end{aligned} (b) \begin{aligned} \lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{|x-1|} &=\lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{-(x-1)}, \quad(|x-1 |=-(x-1) \text { for } x<1) \\ &=\lim _{x \rightarrow 1^{-}}-\sqrt{2 x}\\ &=-\sqrt{2} \end{aligned}
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