# Whether the statement “ If p is a polynomial, then lim x → b p ( x ) = p ( b ) ” is true or false.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2, Problem 7RQ
To determine

## To find: Whether the statement “ If p is a polynomial, then limx→bp(x)=p(b) ” is true or false.

Expert Solution

The statement “If p is a polynomial, then limxbp(x)=p(b) ” is true.

### Explanation of Solution

Given information:

The given statement is “I If p is a polynomial, then limxbp(x)=p(b) ”.

Calculation:

Let consider that p(x) be a polynomial.

Let p(x)=anxn+an1xn1+an2xn2+...+a1x+a0 be the given polynomial of x .

Here, the every power function is continuous. Hence the p(x) is the sum of power functions and continuous everywhere.

The definition of continuity state that a function f(x) is continuous at x=a a if and only if its limits as

xa is f(a) .

Calculate the value limxbp(x) .

limxbp(x)=limxb(anxn+an1xn1+an2xn2+...+a1x+a0)=(limxbanxn+limxban1xn1+limxban2xn2+...+limxba1x+limxba0)=anbn+an1bn1+an2bn2+...+a1b+a0=p(b)

Therefore, the statement “If p is a polynomial, then limxbp(x)=p(b) ” is true.

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