Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 19



Work Step by Step

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}$$ Let $y=mx$; then \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}&=\lim _{x \rightarrow0} \frac{x^{2}-m^{2}x^2}{\sqrt{x^{2}+m^{2}x^2}}\\ &= \lim _{x \rightarrow0} \frac{x^{2}(1-m^{2})}{x\sqrt{1+m^{2}}}\\ &=\lim _{x \rightarrow0} \frac{x(1-m^{2})}{\sqrt{1+m^{2}}}\\ &=0 \end{align*}
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