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Chapter 10: Q. 89 (page 777)

Use limit rules and the continuity of power functions to prove that every polynomial function is continuous everywhere.

Short Answer

Expert verified

The polynomial function f is continuous atx=c.

Step by step solution

01

Step 1. Given Information:

Using limit rules and the continuity of power functions

02

Step 2. Prove:

Consider any polynomial function,

f(x)=anxn+an-1xn-1+⋯+a1x+a0

Assume that c is a real number.

Now by the sum rule, constant multiple rules, and limit of a constant, we have:

limx→cf(x)=anlimx→cxn+an-1limx→cxn-1+⋯+a1limx→cx+a0

03

Step 3. Every polynomial function is continuous everywhere:

Since power functions with positive integer powers are continuous everywhere, we can solve each of the component limits by evaluation, which gives us

limx→cf(x)=ancn+an-1cn-1+⋯+a1c+a0=f(c)

Therefore, the polynomial function f is continuous at x = c.

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Most popular questions from this chapter

In Exercises 22–29 compute the indicated quantities when u=(2,1,−3),v=(4,0,1),andw=(−2,6,5)

role="math" localid="1649400253452" (u×v)×wandu×(v×w)

In Exercises 30–35 compute the indicated quantities when u=(−3,1,−4),v=(2,0,5),andw=(1,3,13)

role="math" localid="1649434688557" v×wandw×v

Find u+vandu-valso sketch u,v,u+v,u-v

role="math" localid="1649595165778" u=1,-2andv=-3,6

In Exercises 30–35 compute the indicated quantities when u=(−3,1,−4),v=(2,0,5),andw=(1,3,13)

Find the volume of the parallelepiped determined by the vectors u, v, and w.

Find PQ→

P=-1,5,4,Q=-1,6,4
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