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Q. 89

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Calculus

Found in: Page 777

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

The polynomial function $f$ is continuous at $x = c$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information:

Using limit rules and the continuity of power functions

Step 2. Prove:

Consider any polynomial function,

$f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$

Assume that c is a real number.

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

$\underset{x \to c}{l i m} f (x) = a_{n} \underset{x \to c}{l i m} x^{n} + a_{n - 1} \underset{x \to c}{l i m} x^{n - 1} + \dots + a_{1} \underset{x \to c}{l i m} x + a_{0}$

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

$\lim_{x \to c} f (x) = a_{n} c^{n} + a_{n - 1} c^{n - 1} + \dots + a_{1} c + a_{0} = f (c)$
Therefore, the polynomial function $f$ is continuous at x = c.

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