# Limit in Multivariable

## Homework Statement

Hi, I have to evaluate the following limit:
$$\lim_{(x,y) \to (1,0)}\frac{x*y-y}{(x-1)^2+y^2}$$

## Homework Equations

I'm pretty sure I have to use the squeeze theorem.

## The Attempt at a Solution

Well, I'm pretty sure it has something to do with the fact that the top factors like this:
$$\lim_{(x,y) \to (1,0)}\frac{y(x-1)}{(x-1)^2+y^2}$$
I'm really new to the squeeze theorem so I don't really know how to use it. I believe I have to find some function comparable to this one that is equal to it or greater than it for all values of x and y and one that is equal or less for all values of x and y. Then I have to prove that both have the same limit, so this function must have it as well.
Oh, and I suspect the limit is 0.
Can someone give me a hand, please?

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LCKurtz
Homework Helper
Gold Member
I would be looking for an argument that the limit doesn't exist.

jbunniii
Homework Helper
Gold Member
You can simplify the problem via a change of variables: let w = x-1. Then the above limit is equivalent to

$$\lim_{(w,y)\rightarrow(0,0)} \frac{wy}{w^2 + y^2}$$

What happens if you let $(w,y) \rightarrow (0,0)$ from different directions?

Doh! Thank you. I graphed it and it appeared that it did exist, but I see now I graphed the wrong thing.