match limit of different paths on a graph

Question

$\lim_{(x,y)\rightarrow 0}\frac{xy^2}{x^2+y^4}$

if we consider the path $x=y^2$ and get $\lim_{(y^2,y)\rightarrow 0}\frac{y^2(y^2)}{(y^2)^2+y^4}=\lim_{y\rightarrow 0}\frac{y^4}{2y^4}=\frac{1}{2}$

but if I used Mathematrica to draw a graph of this function

it looks like the graph doesn't match the limit should be 0.5 when follow the path $x=y^2$,

Answer 1

Since the math is impeccable I would suspect the fault lies with the values of $\Delta x$ and $\Delta y$ used by Mathematica. It has been years since I used Mathematica so I cannot advise you here. But if possible, set $\Delta x$ and $\Delta y$ to smaller values and see if you obtain a similar result.

Answer 2

For $ \lim_{(x,y)\rightarrow (0,0)}\frac{xy^2}{x^2+y^4}, $ if you set $x = -y^2$, the limit is $$ \lim_{y\rightarrow 0}\frac{(-y^2)y^2}{(y^2)^2+y^4} = -\frac 12, $$ and if you set either $x = 0$ or $y = 0$, the limit is $$ \lim_{y\rightarrow 0}\frac{0(y^2)}{0^2+y^4} = \lim_{x\rightarrow 0}\frac{x(0^2)}{x^2+0^4} = 0. $$

Some 3D plotting software, given an arbitrary function over $x$ and $y$, computes the $z$-values at some finite number of $(x,y)$ values in a grid pattern. If the granularity of the grid is $\delta$ units, the software will compute the function at points along the $y$-axis, and it will compute it at points along the lines $x=\delta$ and $x=-\delta$, but it will not compute it at points that are closer than $\delta$ to the $y$-axis while not being exactly on the $y$-axis.

The points that form the "ridges" at heights $z=\frac12$ and $z=-\frac12$ are along the curves $x = y^2$ and $x = -y^2$, which get very close to the $y$ axis while they are still relatively far from the $x$ axis. If the plot is done according to a grid of $(x,y)$ values, inevitably you will "lose" part of the ridges where they get too close to the $y$-axis to show up as plotted values.

It seems a reasonable guess that Mathematica is doing something like that. If you can get it to use a smaller grid, you can make the gap a bit smaller. But in order to make the gap half as large, you must make the grid size one quarter as large.

In general, I think you're likely to have problems like this with any function that is bounded as $(x,y)\to(0,0)$ but that does not have a (path-independent) limit at $(0,0)$.