Like an example, can I say that
$\lim_{(x,y)->(0,0)}\frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}$ = 1
I can substitute $\sqrt{x^2 + y^2}= z$ and then use the Single Variable calculus result. But is doing this step perfectly justified and can we prove that doing this step is actually correct. I know the Multivaraible Taylor series can be got from a single variable Taylor series in this way (that is using such kinds of substitutions). But is there a proof that such a way of substituting is actually correct.
Because in the multivariable limit case the function can tend to zero along infinitely possible paths but when you do this substitution the function will only tend to zero through along two paths ( that is from left or from right of origin).
Similarly how do I calculate multivariable limits in a general case. To show the limit doesn't exist is easy, just show it doesn't exist or is not equal to along 2 paths.
But to actually calculate it is very difficult. You have got infinite paths to check. Is this right or not...
So how do I calculate a multivariable limit in a general case. Is there some epsilon Delta way of calculating the limit or some way of change of coordinates.
And lastly is L' hospital rule valid in multivariable case.