Precalculus (6th Edition) Blitzer

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