anonymous
  • anonymous
Does the limit lim(x,y)-->(0,0) (x^3 + y^3) / (x^2 + y^2) exist? If the limit exists, find its value.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
So, this is a zero over zero limit. Do you know how to deal with these?
anonymous
  • anonymous
in two dimensions, yes.
anonymous
  • anonymous
It's much the same.

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anonymous
  • anonymous
ok?
anonymous
  • anonymous
Not like I haven't tried it already...
anonymous
  • anonymous
Finding it in a text to make sure I don't tell you the wrong thing.
anonymous
  • anonymous
ok
anonymous
  • anonymous
I tried what I would have done it 2 dimensions but still came to 0/0 and then used l'hopital's to get =0 for every try, which would mean it exists... but I am almost certain that it does not.
anonymous
  • anonymous
Well, I can't find it in a text, but I did find this: http://www.physicsforums.com/showthread.php?t=112312
anonymous
  • anonymous
There's a little back and forth about what to do, but they are handling a very similar limit.
anonymous
  • anonymous
alright
anonymous
  • anonymous
Here is a thorough examination: http://www.sinclair.edu/centers/mathlab/pub/findyourcourse/worksheets/calculus/LimitsOfFunctionsOfTwoVariables.pdf
Zarkon
  • Zarkon
\[\left|\frac{x^3 + y^3}{x^2 + y^2}\right|=\left|\frac{(x+y)(x^2+y^2-xy)}{x^2+y^2}\right|=\left|\frac{(x+y)(x^2+y^2)+(x+y)(-xy)}{x^2+y^2}\right|\] \[=\left|\frac{(x+y)(x^2+y^2)}{x^2+y^2}+\frac{(x+y)(-xy)}{x^2+y^2}\right|\] \[=\left|x+y+\frac{(x+y)(-xy)}{x^2+y^2}\right|=\left|x+y-\frac{x^2y+y^2x}{x^2+y^2}\right|\] \[\le |x+y|+\left|\frac{x^2y+y^2x}{x^2+y^2}\right|\le |x+y|+\left|\frac{x^2|y|+y^2|x|}{x^2+y^2}\right|\] \[\le |x+y|+\left|\frac{x^2w+y^2w}{x^2+y^2}\right|\] where \(w=\max\{|x|,|y|\}\) \[=|x+y|+w\to 0\text{ as }(x,y)\to(0,0)\]

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