$$\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$$
(a) Prove that the limit of $f(x, y)$ as $(x, y)$ approaches $(0, 0)$ along any straight line is $0$.
(b) Does $\lim_{(x,y)\to(0,0)} f(x, y)$ exist?
What I'm confused about this question is, for part (b) based on the discounity test the limit clearly does not exist. If we let $x=y^2$ which gives a limit of $0.5$ and if we let $x=y$, the limit approaches $0$. But in part (a), how can the limit approach $0$ when it does not even exist? And another point is that for part (a), we cannot let y=mx to prove that the limit exists along a straight line because that method can only test for discounity, it cannot be used to prove that a limit exists?
Note: what this question is asking is that even though the limits clearly does not exists, we have to prove why it does seem to exists at 0 when we ONLY consider the approach path of
the straight .
Best Answer
In part (a) you only have to show that the limit approaches to 0 if you move along a straight line. This is not a contradiction to your result for $x = y^2$ as $(x, y)$ then moves on a parabola.