Multivariate Calculus - Bruce E. Shapiro

Lecture 10
Limits and Continuity
In this section we will generalize the definition of a limit to functions of more than
one variable. In particular, we will give a meaning to the expression
which is read as
lim f(x, y) = L
(x,y)→(a,b)
“The limit of f(x, y) as the point (x, y) approaches the point (a, b) is L.”
The complication arises from the fact that we can approach the point (a, b) from
any direction. In one dimension, we could either approach from the left or from the
right, and we defined a variety of notations to take this into account:
and then the limit
is defined only when
L + = lim
x→a + f(x)
L − = lim
x→a − f(x)
lim f(x) = L
x→a
L = L + = L − ,
i.e., the limit is only defined when the limits from the left and the right both exist and
are equal to one another. In two dimensions, we can approach from any direction,
not just from the left or the right (figure 10.1).
Let us return to the single-dimensional case, as illustrated in figure 10.2. Whichever
direction we approach a from, we must get the same value. If the function approaches
the same limit from both directions (the left and the right) then we say
the limit exists. Formally, we say that
lim f(x) = L
x→a
exists if and only if for any ɛ > 0, no matter how small, there exists some δ > 0
(that is allowed to depend functionally on ɛ) such that whenever 0 < |x − a| < δ
67