Is it always the case that I can split limits like this when they exist?

Question

I was working on a limit problem thing-a-ma-bob when I happened to wonder the following question:

Is it always the case that $$\lim_{h\to a}f(x(h),y(h))=\lim_{u\to a}\lim_{v\to a}f(x(u),y(v))=\lim_{v\to a}\lim_{u\to a}f(x(u),y(v))$$ when they exist?

That is, I was wondering if having the limit exist is enough to have them all equal. Intuitively, it seems that it should be the case, but I'm not entirely sure about this.