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Limit of a two-variable function

  1. Mar 2, 2017 #1
    1. The problem statement, all variables and given/known data
    Does the following limit exist: ##\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}##?

    2. Relevant equations


    3. The attempt at a solution
    So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting 1. I can't think of any other curves to evaluate on. Does the limit exist or is there another curve that I am not evaluating that gives a different limit?
     
  2. jcsd
  3. Mar 2, 2017 #2

    haruspex

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    If it does exist, it does not matter how many curves you try you will not have proved it. So how about trying to prove it exists?
     
  4. Mar 2, 2017 #3

    Mark44

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    Converting to polar form is fruitful, showing that the limit does exist.
     
  5. Mar 2, 2017 #4
    Converting to polar form, I get, after simplification, ##\displaystyle \lim_{r \rightarrow 0^+} \sqrt{1+\cos \theta \sin^2 \theta}##. I'm not sure how to proceed
     
  6. Mar 2, 2017 #5

    haruspex

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    You seem to have dropped an r.
     
  7. Mar 2, 2017 #6

    Ray Vickson

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    Wrong expression!
     
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