# Limit Problem (Multi)

• Feb 25th 2007, 07:14 PM
fifthrapiers
Limit Problem (Multi)
Evaluate the following limit:

lim([(x,y) -> (0,0)] (xy)/(x^2+y^2)

The limit of (xy)/(x^2+y^2) as (x,y) approaches (0,0).

I have no idea.
• Feb 25th 2007, 07:20 PM
AfterShock
Quote:

Originally Posted by fifthrapiers
Evaluate the following limit:

lim([(x,y) -> (0,0)] (xy)/(x^2+y^2)

The limit of (xy)/(x^2+y^2) as (x,y) approaches (0,0).

I have no idea.

First, consider the lim along the path x = 0. And thus, we have:

lim([(0,y) -> (0,0)] 0/(0 + y^2) = lim(y -> 0) 0 = 0

Then, consider the lim along the path y = 0; thus, we have

lim([(x,0) -> (0,0)] 0/(x^2 + 0) = lim(x -> 0) 0 = 0

Although the limits along the above two paths are the same does not necessarily mean that a limit infact exists. In order for a lim to exist, the lim must be the same along every path through (0,0), not just the two picked. Looking at the path y = x, we have:

lim([(x,x) -> (0,0)] (x*x)/(x^2 + x^2) = lim(x -> 0) x^2/(2x^2) = 1/2

And thus, since the limit along the above path does not match the first two paths, the limit does not exist.
• Feb 25th 2007, 07:41 PM
fifthrapiers
Thanks, one more limit if you have time please.

Consider f(x,y)= 2y^2/(x^2-y^2).

1.) Find the lim along path x = 1. That is, find lim([(1,y)->(1,1)] f(1,y))

My work: lim(y -> 1) [(-2y^2)/(y^2 - 1)] is undefined

2.) Find the lim along path y = 1. That is, find lim([(x,1) -> (1,1)] f(x,1))

Not sure on this bc I think I got 1 wrong.

3.) What can you conclude from the above results?