Calculus (3rd Edition)

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

Chapter 15 - Differentiation in Several Variables - Chapter Review Exercises - Page 835: 12

Answer

$$0$$

Work Step by Step

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}$$ Choose two lines that pass through $(0,0) $ \begin{align*} \frac{y}{x}&=m \end{align*} Let $y=x$; then \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}&=\lim _{x \rightarrow 0} \frac{x^{5} +x^{5} }{x^{4}+x^{4}}\\ &=0 \end{align*} Let $y=2x$; then \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}&=\lim _{x \rightarrow 0} \frac{4x^{5} +8x^{5} }{x^{4}+16x^{4}}\\ &=0 \end{align*} Hence $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}=0$$
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