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: 32



Work Step by Step

Given $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}$$ Consider the line $ y=mx$ that passes through $(0,0)$: \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}&=\lim _{x\to 0 } \frac{mx^2}{\sqrt{x^{2}+m^{2}x^2}}\\ &=\lim _{x\to 0} \frac{mx}{\sqrt{1+m^2}}\\ &=0 \end{align*}
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