Limits & Continuity of Multivariable Functions

Path-dependence of limits and what continuity means in the plane.

The idea

The limit of a function of two variables records the value $f(x, y)$ settles on as its input approaches a point. We write $\lim_{(x,y) \to (a,b)} f(x,y) = L,$ meaning that $f(x, y)$ comes as close to $L$ as we like once $(x, y)$ is close enough to $(a, b)$.

In one variable there are only two ways to approach $a$: from the left and from the right. In the plane there is no such list. The point $(a, b)$ can be approached along either axis, along any line through it, along a parabola, along a spiral — and the limit exists only if every approach yields the same $L$.

Proving and disproving a limit are therefore unequal tasks. To prove one exists we must control every approach at once, and no finite list of paths can do that. To disprove one, two paths suffice: if $f$ settles on different values along them, no single $L$ serves both.

A function $f$ is continuous at $(a, b)$ when the limit there exists and equals $f(a, b)$. When the limit fails to exist, $f$ is not continuous there, whatever value $f(a, b)$ is given.

Ways to work on it

• Walkthrough. Test two paths to show a two-variable limit fails to exist.
• Practice. Evaluate a function along an approach path to the origin.
• Hardest. A limit where all lines agree but a curve disagrees, then continuity.

Not sure where to start? Take the ten-question placement test.