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

[Jadehaan]

Homework Statement

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)=(e^xy-1)/(x²+y²)

Homework Equations

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)=(e^x2-1)/2x²

Thanks for any help.

 

[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?

 

[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.

 

[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

 

[kimanampaka]

http://www.wolframalpha.com/input/?i=%28e^%28x*y%29-1%29%2F%28x%C2%B2%2By%C2%B2%29