Answer
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.
