# Multivariable limits with CAS

1. Mar 15, 2009

### springo

1. The problem statement, all variables and given/known data
I would like to be able to compute multivariable limits with a CAS. I have Mathematica, Derive and an HP 50g.
For example, how would I compute:
$$\lim_{(x,y)\rightarrow(0,0)}\frac{x^2}{y}$$

2. Relevant equations

3. The attempt at a solution
On Mathematica I tried:
Code (Text):
Limit[x^2/y, {x, y} -> {0, 0}]
(doesn't work)
Code (Text):
Limit[x^2/y /. {y -> f[x]}, x -> 0]
(I think this could work, but I need to figure out what f[x] has to be)

2. Mar 15, 2009

### yyat

The limit in your example does not exist (it is undefined), because it depends on how you approach (0,0). For example, f(t,t)->0 as t->0 from the right, but f(sqrt(t),t)->1 as t->0 from the right.

3. Mar 15, 2009

### springo

OK, sorry for choosing a wrong example, I'm just starting multivariable calculus. Anyway for existing limits, is there any way to compute them using a CAS?

4. Mar 15, 2009

### gabbagabbahey

Here's a nice little example using mathematica. Contour Plots are typically a good way to infer whether a limit exists before you go ahead and calculate it.