What are the different forms of substitution to prove limit of multivariable function exists?

Question

I watched a great youtube video about how to prove a limit of a multivariable function exists. It explained that one method is by substitution. For example, we can solve $$lim_{(x, y) \to 0,0} \frac{xy}{\sqrt{x^2 + y^2}}$$

By substituting with polar coordinates - namely letting $x = rcos\theta$ and $y=rsin\theta$

The youtube video mentioned that polar coordinates are not the only accepted form of substitution.

What are other forms of substitution one can use? When do you know which form to substitute with?

Answer 1

Along the line defined by the cartesian equation $y=mx $, the limit is $$\lim_{x\to 0}\frac {mx^2}{|x| \sqrt {1+m^2}} $$ $$=\lim_{x\to 0}\frac {m|x|}{1+m^2}=0$$

it doesn't depend on $m $ so what can you say.

Answer 2

If the function for which you have to compute the limit is defined in $\mathbb{R}^3$ then you can use spherical coordinates.

I think you will find this useful.

It is a generalization of polar coordinates in any dimension.