Limit of a two-variable function

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Homework Statement


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}}##?

Homework Equations




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?
 

Answers and Replies

  • #2
haruspex
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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
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?
 
  • #3
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Homework Statement


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}}##?

Homework Equations




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?
Converting to polar form is fruitful, showing that the limit does exist.
 
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  • #4
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Converting to polar form is fruitful, showing that the limit does exist.
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
 
  • #5
haruspex
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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
You seem to have dropped an r.
 
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  • #6
Ray Vickson
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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

Wrong expression!
 

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