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 - Exercise Set - Page 1160: 2


Yes, the function $ f\left( x \right)=3x+4$ is continuous at $1$.

Work Step by Step

Consider the function $ f\left( x \right)=3x+4$, First check whether the function is defined at the point $ a $ or not. Find the value of $ f\left( x \right)$ at $ a=1$, $\begin{align} & f\left( 1 \right)=3\left( 1 \right)+4 \\ & =3+4 \\ & =7 \end{align}$ The function is defined at the point $ a=1$. Now find the value of $\,\underset{x\to 1}{\mathop{\lim }}\,3x+4\,$, $\begin{align} & \,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right) \\ & =3\left( 1 \right)+4 \\ & =3+4 \\ & =7 \end{align}$ Thus, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7$ Now check whether $\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ or not. From the above, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7\text{ and }f\left( 1 \right)=7$ Therefore, $\,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=f\left( 1 \right)$ Thus, the function satisfies all the properties of being continuous. Hence, the function $ f\left( x \right)=3x+4$ is continuous at $1$.
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