How to find the Limit of two variable using mathematica ?

Question

I was wondering if there is a method to solve the limit of two variable using Mathematica , I try to do this

Limit[(x^4+y^2)/(x^2+y^2),(x,y)->(0,0)]

Limit[(x^4+y^2)/(x^2+y^2),x->0,y->0]

but non of them work ,

Answer 1

While a Mathematica question, it is as much a question about multi-dimensional limits. If you understand what a multi-dimensional limit is, you will know how to use the Limit command to find it. What does $(x,y)\rightarrow(0,0)$ mean? Do you take $x\rightarrow0$ then $y\rightarrow0$? Or the other way around? Or let $y=2x$ and take $x\rightarrow0$?

What this is getting at is that the multi-dimensional limit isn't always well-defined. It can be dependent on which path you take. If you know that the limit exists (which means that every path gives the same limit), then you can choose any path and know that's the limit. Thus if the limit of $f$ is well-defined at (0,0) then the easiest way would be to limit in $x$ then limit in $y$, or

Limit[Limit[f[x,y],x->0],y->0].

However, you have to know that the limit exists for this to be valid! At least check a few other cases paths and make sure they're all the same. If they are not, then the limit does not exist.

So the answer is a bit of math plus computational help.