## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 14

#### Answer

See the proof below.

#### Work Step by Step

Along the $x$-axis $y=0$, the function $\frac{xy}{x^2+y^2}=\frac{0}{x^2+0}=0$ and along the $y$-axis $x=0$, the function $\frac{xy}{x^2+y^2}=\frac{0}{0+y^2}=0$. Now, along the line $y=x$, we have $$\lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}= \lim\limits_{(x,y) \to (0,0)}\frac{xy}{x^2+y^2}\\ =\lim\limits_{(x,y) \to (0,0)}\frac{x^2}{x^2+x^2}\\ =\lim\limits_{(x,y) \to (0,0)}\frac{1}{2}\\ =\dfrac{1}{2}$$ Thus the limits are not the same and hence the overall limit does not exist.

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