Limit of Multivariable Function: x^2+y^2+2xy

[fogel1497]

Homework Statement

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

Homework Equations

^{x = r*cos(theta)
y= r*sin(theta)}

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?

 

[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?

 

[fogel1497]

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

 

[Dick]

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