anonymous
  • anonymous
Go to this site: http://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx Check out example c. In his solution he checks the limit with 3 different approaches. (x,0),(0,y),(x,x), when he evaluates the (x,0) and (0,y) he gets 0/0 but still evaluates it to 0 in the limit where it should be undefined unless (obviously I am) missing something. What am I missing?
Mathematics
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chestercat
  • chestercat
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anonymous
  • anonymous
I don't see an example "c"; I see examples 1 through 6
anonymous
  • anonymous
http://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx Example 1.c, sorry, wrong link!
anonymous
  • anonymous
I think he's right, but leaves something out of the explanation. When y=0 he has this limit: \[\lim_{x\rightarrow0} '{0 \over x^4}\] When x=0 he has this limit: \[\lim_{y\rightarrow0} '{0 \over 3y^4}\] Both of those are constant functions with a single removable singularity (it's technically not a discontinuity because that point is not in the domain). Since it's a constant function, the limit aproaching that point has the same value that the function has at every other point.

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anonymous
  • anonymous
Ok, so how would you say ((x^2)*y)/(x^4+y^2) and ((x^2)*y)/(x^2+y^2) would evaluate to? I'm quite sure the first one is not defined in the point (0,0) but the second one is. What's your approach?
anonymous
  • anonymous
Neither is defined at (0,0), both hit division by zero. I assume you mean how would I evaluate the limits along one axis as the other axis approaches zero, and it's the same as in the example: \[\lim_{(x,0)\rightarrow(0,0)}'{x^2 \cdot y \over x^4+y^2} = \lim_{x \rightarrow 0}'{0 \over x^4} = 0\] \[\lim_{(0,y)\rightarrow(0,0)}'{x^2 \cdot y \over x^4+y^2} = \lim_{y \rightarrow 0}'{0 \over y^2} = 0\] \[\lim_{(x,0)\rightarrow(0,0)}'{x^2 \cdot y \over x^2+y^2} = \lim_{x \rightarrow 0}'{0 \over x^2} = 0\] \[\lim_{(0,y)\rightarrow(0,0)}'{x^2 \cdot y \over x^2+y^2} = \lim_{y \rightarrow 0}'{0 \over y^2} = 0\] As in the example, this isn't enough to show that either limit actually exists, it just fails to show that either limit does not exist.
anonymous
  • anonymous
So, what would one how to do to actually see if either if the limit does exist or does not exist?
anonymous
  • anonymous
As far as I know there is no sure way to prove that the limit does exist; if there is, I doubt it falls within the scope of Calc 3. A quick search turned up http://bit.ly/glqw77 , which may give a sure way; I don't have time to confirm that right now.
anonymous
  • anonymous
Thanks, seems a bit weird that it's so hard to define/solve really.

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