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Limit in Multivariable

  • #1

Homework Statement


Hi, I have to evaluate the following limit:
[tex]\lim_{(x,y) \to (1,0)}\frac{x*y-y}{(x-1)^2+y^2}[/tex]


Homework Equations


I'm pretty sure I have to use the squeeze theorem.


The Attempt at a Solution


Well, I'm pretty sure it has something to do with the fact that the top factors like this:
[tex]\lim_{(x,y) \to (1,0)}\frac{y(x-1)}{(x-1)^2+y^2}[/tex]
I'm really new to the squeeze theorem so I don't really know how to use it. I believe I have to find some function comparable to this one that is equal to it or greater than it for all values of x and y and one that is equal or less for all values of x and y. Then I have to prove that both have the same limit, so this function must have it as well.
Oh, and I suspect the limit is 0.
Can someone give me a hand, please?
 

Answers and Replies

  • #2
LCKurtz
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I would be looking for an argument that the limit doesn't exist.
 
  • #3
jbunniii
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You can simplify the problem via a change of variables: let w = x-1. Then the above limit is equivalent to

[tex]\lim_{(w,y)\rightarrow(0,0)} \frac{wy}{w^2 + y^2}[/tex]

What happens if you let [itex](w,y) \rightarrow (0,0)[/itex] from different directions?
 
  • #4
Doh! Thank you. I graphed it and it appeared that it did exist, but I see now I graphed the wrong thing.
 

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