A multivariable limit asks what happens to a function when all of its input coordinates approach a point together.

For a function of two variables,

$$\lim_{(x,y)\to(a,b)}f(x,y)=L$$

means that $f(x,y)$ approaches $L$ whenever $(x,y)$ approaches $(a,b)$, regardless of the path taken through the plane.

Many possible paths

In one variable, a point can be approached essentially from the left or the right. In two variables there are infinitely many paths.

Consider

$$f(x,y)=\frac{x^2-y^2}{x^2+y^2}.$$

Along $y=0$,

$$f(x,0)=1,$$

while along $x=0$,

$$f(0,y)=-1.$$

Because two paths give different limiting values,

$$\lim_{(x,y)\to(0,0)}f(x,y)$$

does not exist.

Matching paths are not enough

Finding several paths that give the same value cannot prove a multivariable limit, because another path may behave differently. Path tests are therefore powerful for disproving limits but not generally for proving them.

Proving a limit

A limit can be established by algebraic simplification, comparison estimates or known continuity. For example,

$$|xy|\le\frac{x^2+y^2}{2},$$

so as $(x,y)\to(0,0)$,

$$|xy|\to0,$$

which proves

$$\lim_{(x,y)\to(0,0)}xy=0.$$

Multivariable limits extend the idea of nearby behavior from a line to spaces with many possible directions of approach.