# Limit as 9x,y) approaches (0,0)

1. Oct 19, 2009

1. The problem statement, all variables and given/known data

Justify if the limit of the following function exists as (x,y) approaches (0,0). If it exists find the limit using the squeezing technique.
f(x,y)=(exy-1)/(x2+y2)

2. Relevant equations

3. The attempt at a solution

I found the limit of f(x,0) to approach 0
I found the limit of f(0,y) to approach 0
Since this is insufficient I found the limit of f(x,x)=(ex2-1)/2x2

Thanks for any help.

2. Oct 19, 2009

### lanedance

why not extend the evaluation to all linear trajectories towards the origin given by y = cx, for some real c

then evaluate the limit of
(lim x->0) g(x) = f(x,cx)
if it is indeterminate, try using L'hopitals rule

does the limit depend on c?

3. Oct 20, 2009

### HallsofIvy

Unfortunately, even if the limit, as you approach (0,0) along every line is the same, it might still be different for some curve- and so the limit might not exist.

4. Oct 20, 2009

### lanedance

ok, but if you can show that dependent on the line of approach, the result differs, you have shown the limit does not exist

5. Nov 2, 2009