   Chapter 2.3, Problem 57E

Chapter
Section
Textbook Problem

If p is a polynomial, Show that limx→a p(x) = p(a)

To determine

To show: If p is a polynomial, then limxap(x)=p(a).

Explanation

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exists, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Proof:

Let p be a polynomial.

So take p(x)=α0+α1x+α2x2++αnxn where {αi:i(1,2,,n)} are constants.

The limit of the polynomial p(x) as x approaches a.

limxap(x)=limxa(α0+α1x+α2x2++αnxn)=limxaα0+limxaα1x+limxaα2x2++limxaαnxn[By limit law 1]=limxaα0+α1limxax+α2limxax2++αnlimxaxn[By limit law 3]=α0+α1limxax+α2

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