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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+鈰�+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鈫�cf(x)=anlimx鈫�cxn+an-1limx鈫�cxn-1+鈰�+a1limx鈫�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鈫�cf(x)=ancn+an-1cn-1+鈰�+a1c+a0=f(c)

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

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