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Expert-verified Found in: Page 777 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Use limit rules and the continuity of power functions to prove that every polynomial function is continuous everywhere.

The polynomial function $\mathbit{f}$ is continuous at $\mathbit{x}\mathbf{=}\mathbit{c}$.

See the step by step solution

## Step 1. Given Information:

Using limit rules and the continuity of power functions

## Step 2. Prove:

Consider any polynomial function,

$\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}{\mathbit{a}}_{\mathbf{n}}{\mathbit{x}}^{\mathbf{n}}\mathbf{+}{\mathbit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathbit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathbit{a}}_{\mathbf{1}}\mathbit{x}\mathbf{+}{\mathbit{a}}_{\mathbf{0}}$

Assume that c is a real number.

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

$\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}{\mathbf{a}}_{\mathbf{n}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathbit{x}}^{\mathbf{n}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathbit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbit{x}\mathbf{+}{\mathbit{a}}_{\mathbf{0}}$

## 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

$\underset{\mathbf{x}\mathbf{\to }\mathbf{c}}{\mathbf{lim}}\mathbit{f}\mathbf{\left(}\mathbit{x}\mathbf{\right)}\mathbf{=}{\mathbit{a}}_{\mathbf{n}}{\mathbit{c}}^{\mathbf{n}}\mathbf{+}{\mathbit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathbit{c}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathbit{a}}_{\mathbf{1}}\mathbit{c}\mathbf{+}{\mathbit{a}}_{\mathbf{0}}\phantom{\rule{0ex}{0ex}}\mathbf{=}\mathbit{f}\mathbf{\left(}\mathbit{c}\mathbf{\right)}$
Therefore, the polynomial function $\mathbit{f}$ is continuous at x = c. ### Want to see more solutions like these? 