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).

Evaluate the limit along the line y=mx where m∈R.

f(x,y)
=(x+y)/(y^2-x^2)
=(x+mx)/(m^2x^2-x^2)
=x(1+m)/[x^2(m+1)(m-1)]
=1/[(m-1)x]

Since the limit depends on m, the limit does not exist.

To find the limit of the given function as (x, y) approaches the point (-1, 1), we need to consider the behavior of the function as we approach this point from different paths.

Let's start by simplifying the given function:

f(x, y) = (x + y) / (y^2 - x^2)

Next, we can substitute the point (-1, 1) into the function and see what value the function takes at that point:
f(-1, 1) = (-1 + 1) / (1^2 - (-1)^2) = 0 / 0

Here we encounter an indeterminate form of 0/0, which means that we can't determine the value of the function at (-1, 1) just by direct substitution. To determine if the limit exists, we need to consider the limit of the function as (x, y) approaches (-1, 1) along different paths.

Let's analyze the limit along two different paths: the x-axis (y = 0) and the line y = x.

1. Along the x-axis (y = 0):
Substituting y = 0 in the function gives:
f(x, 0) = (x + 0) / (0^2 - x^2) = x / (-x^2) = -1 / x

As x approaches -1 along the x-axis, we have:
lim of f(x, 0) as x approaches -1 = lim of (-1 / x) as x approaches -1 from the x-axis = -1 / (-1) = 1

2. Along the line y = x:
Substituting y = x in the function gives:
f(x, x) = (x + x) / (x^2 - x^2) = 2x / 0

Here, we encounter another indeterminate form of 2x/0. By simplifying, we can see that the function approaches infinity as x approaches (-1).

Now, since the limit along the x-axis (y = 0) approaches 1 and along the line y = x, it approaches infinity, we can conclude that the limit of the function as (x, y) approaches (-1, 1) does not exist.