Chapter 13 - Partial Derivatives - 13.2 Limits And Continuity - Exercises Set 13.2 - Page 926: 39

Limit does not exit

We have to find the limit \[ \begin{aligned} & f(x, y)=\frac{x^{2}}{x^{2}+y^{2}} \\ &=\lim _{(x, y) \rightarrow(0,0)} f(x, y) \\ &=\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}} \\ &=\lim _{x \rightarrow 0} \frac{1}{1+\frac{y^{2}}{x^{2}}} \\ &=\text { not defined } \end{aligned} \] So, the discontinuity can't be removed.