Thread: 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.

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.

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?