Calculus 10th Edition

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

Chapter 13 - Functions of Several Variables - 13.2 Exercises - Page 887: 28

Answer

$$\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1}=2 $$

Work Step by Step

Given$$\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1} $$ So, by multiplying the numerator and denominator by the conjugate of the denominator, we get \begin{align} L&=\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1} \\ &=\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1} \cdot \frac{\sqrt{x-y}+1}{\sqrt{x-y}+1} \\ &=\lim _{(x, y) \rightarrow(2,1)} \frac{(x-y-1)(\sqrt{x-y}+1)}{(\sqrt{x-y})^2-1^2} \\ &=\lim _{(x, y) \rightarrow(2,1)} \frac{(x-y-1)(\sqrt{x-y}+1)}{(x-y-1)} \\ &=\lim _{(x, y) \rightarrow(2,1)}(\sqrt{x-y}+1)\\ &= \sqrt{2-1}+1\\ &=2 \end{align}
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