## Calculus (3rd Edition)

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.