Multivariable Limits

  • Thread starter fogel1497
  • Start date
  • #1
12
0

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
Science Advisor
Homework Helper
26,260
619
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
12
0
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
Science Advisor
Homework Helper
26,260
619
If you get different answers for different values of theta, yes, the limit does not exist.
 

Related Threads on Multivariable Limits

  • Last Post
Replies
1
Views
987
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
982
  • Last Post
Replies
3
Views
889
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
10
Views
795
  • Last Post
Replies
13
Views
1K
  • Last Post
Replies
3
Views
982
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
5
Views
2K
Top