Multivariable Limits

  • Thread starter fogel1497
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  • #1
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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?
 

Answers and Replies

  • #2
Dick
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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
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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
Dick
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If you get different answers for different values of theta, yes, the limit does not exist.
 

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