Htykuut

$lim_{(x,y)to(0,0)}frac{x^2y^2}{sin(x)cos(y)}$ is it allowed to split a multi-variable limit into its component variables as in the next step?

$= (lim_{xto0}frac{x^2}{sin(x)})(lim_{xto0}frac{y^2}{sin(y)})$ this is an indeterminate form and now I use L'Hopital

$=(lim_{xto0}frac{2x}{cos(x)})(lim_{xto0}frac{2y}{cos(y)})$

$=(frac{0}{1})(frac{0}{1})=0$

Even switching the order of taking a limit can lead to problems.

As an example: $ limlimits_{m to infty} limlimits_{n to infty} {frac{1}{n}} ^{frac{1}{m}} = limlimits_{m to infty} 0 ^{frac{1}{m}} = 0 neq 1 = limlimits_{n to infty} 1 = limlimits_{n to infty} limlimits_{m to infty} frac{1}{n} ^frac{1}{m} $

While: $ limlimits_{(m,n) to (infty, infty)} {frac{1}{n}} ^{frac{1}{m}}$ does not exist.

At this point I would like to refer to this post well written post and hope this answers some further questions of yours: https://math.stackexchange.com/q/15257

No, it is generally not allowed as it may be false. But you can do as follows:

$$lim_{(x,y)to(0,0)};xcdotfrac x{sin x}cdotfrac{y^2}{cos y}=0cdot1cdotfrac01=0$$

It is actually easier:
$$
left|frac{x^2 y^2}{sin x cos y}right| leq 2left|frac{x}{sin x} right| left| y^2 right| left| x right| leq
|xy^2| =o(1)
$$
as $(x,y) to (0,0)$.