$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
The potential difficulty is largely due to the fact that there are many ways to "approach'' a point in the $$x$$-$$y$$ plane. If we want to say that $$\lim_{(x,y)\to(a,b)}f(x,y)=L$$, we need to capture the idea that as $$(x,y)$$ gets close to $$(a,b)$$ then $$f(x,y)$$ gets close to $$L$$. For functions of one variable, $$f(x)$$, there are only two ways that $$x$$ can approach $$a$$: from the left or right. But there are an infinite number of ways to approach $$(a,b)$$: along any one of an infinite number of lines, or an infinite number of parabolas, or an infinite number of sine curves, and so on. We might hope that it's really not so bad---suppose, for example, that along every possible line through $$(a,b)$$ the value of $$f(x,y)$$ gets close to $$L$$; surely this means that "$$f(x,y)$$ approaches $$L$$ as $$(x,y)$$ approaches $$(a,b)$$''. Sadly, no.