I am familiar with the squeeze law being used for a single-variable limit context, but was wondering if it can be applied to the following multivariable limit too: $$\lim_{(x,y)\to(0,0)}\frac{x^2y^3\ln(1+x)}{x^4+y^6}$$
that is, from AM-GM: $$0<\left| \frac{x^2y^3\ln(1+x)}{x^4+y^6} \right|≤\frac{{|x^2y^3|}|\ln(1+x)|}{2{|x^2y^3|}}=\frac{1}{2}\ln(1+x).$$
then I basically just considered this in a single-variable context by shrinking $x\to 0$ and applying the squeeze law: $\frac{1}{2}\ln(1+x)\to 0$. So the limit is $0$. I am not sure if transforming the multivariable limit to an equivalent single variable one is valid or not.
Moreover, suppose I couldn't simplify the inequality above (assuming I did it correctly) or found a different inequality with bounds in terms of both $x,y$, is there a multivariable version of the central limit that I can directly apply?