Find the limit, if it exists, or show that the limit does not exist. lim of x+y/y^2-x^2 as (x,y) approaches the point (-1,1).

To evaluate the limit of a function as a point is approached, we can use the standard limit laws and techniques. Let's break down the given function and determine if the limit exists.

We need to find the limit of (x+y)/(y^2-x^2) as (x,y) approaches the point (-1,1).

First, let's examine the denominator, y^2 - x^2. We notice that the denominator can be factored using the difference of squares as (y+x)(y-x).

Now, let's rewrite the function using the factored form:
(x+y)/[(y+x)(y-x)]

Next, we consider the limit by approaching the point (-1,1). This means we want to evaluate the function values as (x,y) gets close to (-1,1), but not equal to it.

Let's evaluate the limit by substituting the given point (-1,1) into the function:
((-1)+1)/[(1+(-1))(1-(-1))]

Simplifying further:
0/[(1+1)(1+1)]
0/[2*2]
0/4 = 0

Therefore, the function's limit as (x,y) approaches (-1,1) exists and is equal to 0.