Prove-that-those-functions-below-don-t-have-limit-a-lim-x-y-0-0-xy-x-2-y-2-b-lim-x-y-0-0-xy-y-3-x-2-y-2-

Question Number 11395 by Joel576 last updated on 23/Mar/17

Prove that those functions below {don}^{'} t have limit

a) \lim_{(x, y) \to (0, 0)} \frac{x y}{x^{2} + y^{2}}

b) \lim_{(x, y) \to (0, 0)} \frac{x y + y^{3}}{x^{2} + y^{2}}

Commented by prakash jain last updated on 23/Mar/17

a)

x = r \cos θ

y = r \sin θ

\frac{x y}{x^{2} + y^{2}} = \frac{r^{2} \cos θ \sin θ}{r^{2}} = \frac{\sin 2 θ}{2}

\lim_{(x, y) \to (0, 0)} \frac{x y}{x^{2} + y^{2}} = \lim_{r \to 0} \frac{\sin 2 θ}{2} = \frac{\sin 2 θ}{2}

The limit value depends on the angle of

the line chosen to approach (0, 0) so the

limit does not exist .