# Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 13

The limit does not exist.

#### Work Step by Step

The limit $$\lim\limits_{(x,y) \to (0,0)}\frac{y^2}{x^2+y^2}$$ does not exist. If one considers the paths along the lines $y=mx$, we get $$\lim\limits_{(x,y) \to (0,0)}\frac{y^2}{x^2+y^2}=\frac{m^2}{1+m^2}$$ which depends on the slope $m$ of the line and hence the limit does not exist.

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