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: 11


Does not exist

Work Step by Step

Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y+x y^{2}}{x^{2}+y^{2}}$$ 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 y+x y^{2}}{x^{2}+y^{2}}&=\lim _{x \rightarrow 0} \frac{x^2+x^{3}}{x^{2}+x^{2}}\\ &=\frac{1}{2} \end{align*} Let $y=2x$; then \begin{align*} \lim _{(x, y) \rightarrow(0,0)} \frac{x y+x y^{2}}{x^{2}+y^{2}}&=\lim _{x \rightarrow 0} \frac{2x ^2+4x ^{3}}{x^{2}+4x^{2}}\\ &=\frac{2}{5} \end{align*} Since the two limits are different, then the given limit does not exist.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.