• Support PF! Buy your school textbooks, materials and every day products Here!

Limit of a two variable function

  • Thread starter Jalo
  • Start date
  • #1
120
0

Homework Statement


Hi.
I'm trying to solve the limit of the function:

lim (x,y)→(0,0)f(x,y) =
= lim (x,y)→(0,0) 3 * x2 / (x2 + y2)


Homework Equations





The Attempt at a Solution



x2 / (x2 + y2) is a limited function. Its image is always between 0 and 1. However I don't know what to do from here on.

What I tought was that since (x,y) → (0,0) then I can say that x = y and
x2 / (x2 + y2)
becomes 1/2. I don't think I can do this tho...

Any help or hint will be highly appreciated.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,770
911
Unfortunately, because of the extra dimension, limits in R2 are much more difficult than in R. In R, you only have to prove that the limits "from above" and "from below" are equal. In R2, the limit as you approach the point on any path, even complicated curves, are the same.

Since you can't look at every possible curve, it is much easier to prove that a limit does NOT exist by showing that there exist different paths that give different limits. For example, IF you approach (0, 0) along the path y= x, you get [itex]x^2/(x^2+ y^2)= x^2/2x^2= 1/2[/itex] for all x so the limit is 1/2. Now, suppose you approach (0, 0) along the path (x, 0), the x-axis. What limit do you get? Although that should be enough, you could also look at the limit approaching (0, 0) along the line (0, y).
 
  • #3
52
0
What are the limits when you let x vary but hold y constant and when you let y vary but hold x constant?

*EDIT* Woops -- just saw that HallsofIvy beat me to the response.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,770
911
Nyah- Nyah!!
 
  • #5
120
0
Thanks! That was very helpful!

D.
 

Related Threads for: Limit of a two variable function

  • Last Post
Replies
5
Views
624
  • Last Post
Replies
2
Views
934
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
8
Views
1K
Replies
8
Views
2K
Replies
10
Views
2K
Replies
1
Views
5K
Top