Chapter 13 - Partial Derivatives - 13.2 Limits And Continuity - Exercises Set 13.2 - Page 925: 16

Limit does not exist

We have to find the limit \[ =\lim _{(x, y) \rightarrow(0,0)} \frac{1-x^{2}-y^{2}}{x^{2}+y^{2}} \] Let $z=x^{2}+y^{2}$. So if, $(x, y) \rightarrow(0,0) \Rightarrow z \rightarrow 0^{+}$ \[ \begin{array}{l} =\lim _{z \rightarrow 0^{+}} \frac{(1-z)}{z} \\ =\frac{1}{0^{+}} \\ =\infty \\ =\text { Not defined } \end{array} \]