Limit of a Function of Two Variables

  • Thread starter Mindstein
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  • #1
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Homework Statement


Evaluate the limit of the function f(x,y) = [tex]\frac{y^3}{x^2+y^2}[/tex]


Homework Equations





The Attempt at a Solution


Well, I approached this problem using the multiple-path method and found the following:

[tex]\stackrel{lim}{x\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y
[tex]\stackrel{lim}{y\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = 0

and am having trouble interpreting these results. I tried doing a polar substitution and found that:

[tex]\stackrel{lim}{r\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y*sin2(theta)

My calculus book is very short on the topic, I am pretty much left in the dark. Please bring me into the light. haha.
 

Answers and Replies

  • #2
1,101
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Are you trying determine the limit of the function at the origin? Try using the fact that [itex]x^2 + y^2 \geq y^2[/itex] and apply the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken]
 
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  • #3
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Are you trying determine the limit of the function at the origin? Try using the fact that [itex]x^2 + y^2 \geq y^2[/itex] and apply the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken]

Yes, I am trying to determine the limit as x,y approaches the origin. The problem is that I have no clue how to apply the squeeze theorem. I cant find any good sites on it.
 
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  • #4
statdad
Homework Helper
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google squeeze theorem - it's also called the sandwich theorem, or the pinching theorem
 
  • #5
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Ahhh okay, so what I would want to say is that [tex]y^2[/tex] [tex]\leq[/tex][tex]\frac{y^3}{x^2+y^2}[/tex][tex]\leq[/tex]x2+y2 ?
 

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