Suggested languages for you:

Americas

Europe

Q. 89

Expert-verifiedFound in: Page 777

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 $\mathit{f}$ is continuous at $\mathit{x}\mathbf{=}\mathit{c}$.

Using limit rules and the continuity of power functions

Consider any polynomial function,

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

Assume that c is a real number.

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

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

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

$\underset{\mathbf{x}\mathbf{\to}\mathbf{c}}{\mathbf{lim}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{a}}_{\mathbf{n}}{\mathit{c}}^{\mathbf{n}}\mathbf{+}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{c}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}\mathbf{\cdots}\mathbf{+}{\mathit{a}}_{\mathbf{1}}\mathit{c}\mathbf{+}{\mathit{a}}_{\mathbf{0}}\phantom{\rule{0ex}{0ex}}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{c}\mathbf{\right)}$

Therefore, the polynomial function $\mathit{f}$ is continuous at x = c.

94% of StudySmarter users get better grades.

Sign up for free