Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 11 - Section 11.3 - Limits and Continuity - Concept and Vocabulary Check - Page 1160: 1


The complete statement is, “A function f is continuous at a when three conditions are satisfied. (a) f is defined at a, so that $ f\left( a \right)$ is a real number. (b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. (c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ $=$ $ f\left( a \right)$ ”

Work Step by Step

For a function to be continuous at a point a, the function must satisfy the following three conditions: (a) f is defined at a. (b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. (c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ For example, consider a function $ f\left( x \right)=2x $ To check whether the function is continuous at the point $ a=3$ or not, Find the value of $ f\left( x \right)$ at $ a=3$, $ f\left( 3 \right)=2\left( 3 \right)=6$ The function is defined at the point $ a=3$. Now find the value of $\,\underset{x\to 3}{\mathop{\lim }}\,2x $, $\begin{align} & \,\underset{x\to 3}{\mathop{\lim }}\,2x=2\underset{x\to 3}{\mathop{\lim }}\,x \\ & =2\left( 3 \right) \\ & =6 \end{align}$ Thus, $\,\underset{x\to 3}{\mathop{\lim }}\,2x=6$ From the above two steps, $\,\underset{x\to 3}{\mathop{\lim }}\,2x=6=f\left( 3 \right)$. Thus, the function satisfies all the conditions of being continuous. Hence, the function $ f\left( x \right)=2x $ is continuous at $3$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.