# calculus and analysis – question on correct use of Limit for multivariable function

V 12.1 on windows.

This limit $$lim_{(x rightarrow 0,yrightarrow 0)} frac{x^2-y^2}{x^2+y^2}$$ depends on the direction. So the limit does not exist, or could be written as Maple does it, which is $$-1dots1$$, here is the help from Maple on this:

How can one get Mathematica to give this result? Now Mathematica says the limit is $$1$$. I tried the `Direction` option but not able to make it change its mind.

``````f = (x^2 - y^2)/(x^2 + y^2);
Limit(f, {x -> 0, y -> 0})
(* 1 *)
``````

But we see the limit depends on the direction

`````` Limit(Limit(f, x -> 0), y -> 0)
(* -1 *)

Limit(Limit(f, y -> 0), x -> 0)
(*  1 *)
``````

Here is also Maple to confirm

``````restart;
f:=(x^2-y^2)/(x^2+y^2);
limit(f, (x=0,y=0));
``````

Btw, this is not the only one I found, here is another

``````f = (x^2*y^2)/(x^4 + y^4);
Limit(f, {x -> 0, y -> 0})
(* 0 *)
``````

Maple gives

``````restart;
f:=x^2*y^2/(x^4+y^4);
limit(f,(y=0,x=0))
(* 0 .. 1/2 *)
``````

And another one (this one is from youtube actually, so you can see they also say there the limit does not exist)

``````f = (x^4 - 4 y^2)/(x^2 + 2 y^2);
Limit(f, {x -> 0, y -> 0})
(* 0 *)

restart;
f:=(x^4-4*y^2)/(x^2+2*y^2);
limit(f, (x=0,y=0));
(* -2 .. 0 *)
``````

So I have feeling I am not using Limit in Mathematica correctly, or missing something about its correct use, but do not now know how to correct it. As I said, I tried different `Direction` option.