254 CHAPTER 14. VECTOR VALUED FUNCTIONS OF ONE VARIABLE

One can also consider limits as a variable “approaches” infinity. Of course nothing is“close” to infinity and so this requires a slightly different definition.

limt→∞

f (t) =L

if for every ε > 0 there exists l such that whenever t > l,

|f (t)−L|< ε (14.1)

andlim

t→−∞f (t) =L

if for every ε > 0 there exists l such that whenever t < l, (14.1) holds.

Note that in all of this the definitions are identical to the case of scalar valued functions.The only difference is that here |·| refers to the norm or length in Rp where maybe p > 1.

Example 14.1.2 Let f (t) =(cos t,sin t, t2 +1, ln(t)

). Find limt→π/2f (t) .

Use Theorem 12.5.5 on Page 223 and the continuity of the functions to write this limitequals (

limt→π/2

cos t, limt→π/2

sin t, limt→π/2

(t2 +1

), limt→π/2

ln(t))

=

(0,1, ln

(π2

4+1), ln(

π

2

)).

Example 14.1.3 Let f (t) =( sin t

t , t2, t +1). Find limt→0f (t).

Recall that limt→0sin t

t = 1. Then from Theorem 12.5.5 on Page 223, limt→0f (t) =(1,0,1).

14.2 The Derivative And IntegralThe following definition is on the derivative and integral of a vector valued function of onevariable.

Definition 14.2.1 The derivative of a function f ′ (t), is defined as the following limit when-ever the limit exists. If the limit does not exist, then neither does f ′ (t).

limh→0

f (t +h)−f (t)h

≡ f ′ (t)

As before,

f ′ (t) = lims→t

f (s)−f (t)s− t

.

The function of h on the left is called the difference quotient just as it was for a scalarvalued function. If f (t) = ( f1 (t) , · · · , fp (t)) and

∫ ba fi (t) dt exists for each i = 1, · · · , p,

then∫ b

a f (t) dt is defined as the vector(∫ b

af1 (t) dt, · · · ,

∫ b

afp (t) dt

).

This is what is meant by saying f ∈ R([a,b]).