 
Summary: Curves in Rn
1. Limits, continuity and differentiation.
Throughout this section, I is an interval, a I and
r = (r1, . . . , rn) : I Rn
.
In 12.5 in the book n = 3. Often, for the sake of brevity, we will say r is a curve.
Notice the difference between the range of r (which the book calls the track) and r
itself.
Definition 1.1. Suppose l = (l1, . . . , ln) Rn
. We say r(t) approaches l as t
approaches a and write
(1) lim
ta
r(t) = l
if for each > 0 there is > 0 such that
t I and 0 < t  a < r(t)  l < .
Theorem 1.1. Suppose l = (l1, . . . , ln) Rn
. Then (1) holds if and only if
lim
ta
