Changing variables in limits (looking for the right theorem)

Asked Oct 30 '20 at 16:36

Active Oct 30 '20 at 16:36

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Im looking for the right thorem that states under which conditions we can change the variables when calculating limits of 2 variables to polar coordinates variables. All I found was way more complicates theorems with nore variables involving del operators like divergence and stuff. I'll be glad if someone can point me to this theorem, preferable with proof.

Thanks in advance

asked Oct 30 '20 at 16:36

FreeZe

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Does this answer your question? Finding a limit using change of variable- how come it works? – Eminem Oct 30 '20 at 16:39
No actually I mentioned I’m looking for generalization for 2 variables function – FreeZe Oct 30 '20 at 17:01
polar coordinates give the correct limit when you have a homogeneous expression, either a polynomial with all degrees equal, terms of the form $x^j y^{n-j}$ with fixed $n,$ or the quotient of two such polynomials, while different degree on numerator $n_N$ and denominator $n_D$ is allowed – Will Jagy Oct 30 '20 at 17:20
The easiest example of failure is $ \frac{2x^2y}{x^4 + y^2} $ as we get different limits along the curve $y=x^2$ from $y = -x^2 $ – Will Jagy Oct 30 '20 at 17:25

Changing variables in limits (looking for the right theorem)

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