Precalculus (6th Edition) Blitzer

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

Chapter 11 - Section 11.1 - Finding Limits Using Tables and Graphs - Exercise Set - Page 1139: 33


The value of the limit notation is, $\underset{x\to 3}{\mathop{\lim }}\,2x+1=7$.

Work Step by Step

Consider the provided function, $ f\left( x \right)=2x+1$. This equation $ f\left( x \right)=2x+1$ is in the form of $ y=mx+b $, where ‘ $ m $ ’ is the slope and ‘ $ b $ ’ is the y-intercept. Compare the function $ f\left( x \right)=2x+1$ with $ y=mx+b $ Hence, here the slope $\left( m \right)$ is 2 and the y-intercept $\left( b \right)$ is 1. To graph the function $ f\left( x \right)=2x+1$ using the slope and y-intercept, follow the following steps. Step 1: plot the y-intercept $\left( 0,b \right)$ which is the point $\left( 0,1 \right)$. This point is always lying on the vertical axis, which is y axis. Step 2: From the y-intercept plot another point using the slope. Here, the slope is $ m=\frac{y}{x}=2$ that means from the y-intercept move 2 units up and 1 unit right. Now consider the provided limit, $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$, where $ f\left( x \right)=2x+1$. Consider the obtained graph of $ f\left( x \right)=2x+1$ To find $\underset{x\to 3}{\mathop{\lim }}\,2x+1$, examine the portion of the graph near $ x=3$. As x gets closer to 3, the value of $ f\left( x \right)$ gets closer to the y-coordinate of 7. This point $\left( 3,7 \right)$ is as shown in the above graph. The point $\left( 3,7 \right)$ has a y-coordinate of 7. Thus, $\underset{x\to 3}{\mathop{\lim }}\,2x+1=7$. Hence the value of $\underset{x\to 3}{\mathop{\lim }}\,2x+1$ is 7.
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