Multivariate Calculus - Bruce E. Shapiro

LECTURE 10. LIMITS AND CONTINUITY 69
then |f(x) − L| < ɛ. No matter how close we get to L along the y-axis (within a
distance ɛ), there is some interval about a along the x-axis (of width we call 2δ such
that, if we draw a box about the point (a, L) of size δ ×ɛ, and then slowly shrink the
box to zero, the function will always remain within the box. If the limit does not
exist, then as we shrink the box, at some point there will not be a representation of
the function from left to right across the box (see figure 10.3).
Figure 10.3: Limits of a function of a single variable. On the left, the limit exists,
and we can shrink a box about (a, L) however small we like, and the function remains
entirely within the box. On the right, we eventually get to a point where the function
from the left is not in the box, and the limit does not exist.
Now consider the case of a limit in three dimensions. Rather than approaching
the point to within an interval of width 2δ and letting the size δ → 0 we can approach
in the xy plane from any direction. So the interval becomes a disk of radius δ, and
the box becomes a cylinder of radius δ and height ɛ.
Definition 10.1 A neighborhood of radius δ of a point P is the set of all points
Q satisfying ‖P − Q‖ < δ. In 1D, a neighborhood is called an interval. In 2D, a
neighborhood is called a disk. In 3D (and higher) a neighborhood of P is called a
ball.
Mathematically, the definition of a limit is essentially the same in all dimensions.
Definition 10.2 Let f(x, y) be a function of two variables.
f(x, y) as (x, y) approaches (a, b), which we write as
Then the limit of
lim f(x, y) = L
(x,y)→(a,b)
exists if and only if for any ɛ > 0 there exists some δ > 0 such that whenever
0 < ‖f(x, y) − (a, b)‖ < δ (i.e., (x, y) is in a neighborhood of radius δ of (a, b)) then
|f(x, y) − L| < ɛ (i.e., f(x, y) is in some neighborhood of radius ɛ of L.
Math 250, Fall 2006 Revised December 6, 2006.