Multivariable Limits

1. May 10, 2009

fogel1497

1. The problem statement, all variables and given/known data

Find the limit of:
(x^2+y^2+2xy)/(x^2+y^2)

2. Relevant equations
x = r*cos(theta)
y= r*sin(theta)

3. The attempt at a solution

So what I did was change to polar coordinates. Then it simplifies to:

(r^2 + 2r^2cos(theta)sin(theta) )/r^2

Factoring out an r^2 from everything you get:
1 + 2cos(theta)sin(theta)

And the limit as r and theta go to zero would appear to be 1. However if I plug in numbers very close to zero on my calculator it tells me that the limit is 2. Little help?

2. May 10, 2009

Dick

Your limit depends on theta. That means the limit is different as you approach (0,0) in different ways. E.g. if you approach along the x axis, x=t, y=0 (theta=0), you get 1. If you approach along a line x=t, y=t, (theta=pi/4) you get 2. What does this tell you about the existence of the limit?

3. May 10, 2009

fogel1497

It doesnt exist? So when i do that method if I find i have thetas in my answer then the limit does not exist?

4. May 10, 2009

Dick

If you get different answers for different values of theta, yes, the limit does not exist.