At first I thought that the limit existed, but since the degree of the numerator is less than the degree of the denominator, that gives a hint as to the nonexistence of the limit. As such, the squeeze theorem cannot be applied and I have to find 2 paths where the limits give different values. I've tried many ways but they all end up giving 0.
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2Hint: Choose a path that approaches the origin such that the denominator collapses into one term. – Sammy Black Apr 21 '22 at 03:08
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2@SammyBlack you fell into the trap, that proves nothing – Ninad Munshi Apr 21 '22 at 03:50
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1see https://math.stackexchange.com/questions/4141375/help-in-final-steps-showing-lim-x-y-rightarrow-0-0-frac-leftx-righta/4146090#4146090 – Will Jagy Apr 21 '22 at 03:57
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We can bound the expression
$$0 \leq \frac{|x|y^2}{|x|^5+y^2} \leq \frac{|x|y^2}{y^2} = |x|$$
Thus the limit exists and equals $0$ by squeeze theorem.
Ninad Munshi
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