Chapter 11, Problem 11RE

To determine

To Calculate:

The of given function by direct substitution.

Answer to Problem 11RE

The limit of $f\left(x\right)$ is $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)=5$ .

Explanation of Solution

Given Information:

  $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)$

Calculation:

Consider the following limit

  $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)$

To get the limit of the function by using the direct substitution method, replace the value of $x$ by the defined value:

  $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)=\underset{x\to 4}{\lim }\frac{1}{2}x+\underset{x\to 4}{\lim }3$

Use the property of limit $\underset{x\to 4}{\lim }b=b$

  $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)=\underset{x\to 4}{\lim }\frac{1}{2}x+3$

Use the properties of limit $\underset{x\to 0}{\lim }\left[f\left(x\right)\pm g\left(x\right)\right]=\underset{x\to 0}{\lim }f\left(x\right)\pm \underset{x\to 0}{\lim }g\left(x\right)$ and

  $\underset{x\to 0}{\lim }kf\left(x\right)=k\underset{x\to 0}{\lim }f\left(x\right)$

Therefore,

  $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)=\frac{1}{2}\underset{x\to 4}{\lim }x+3$

Replace $x$ by $4$ ,

  $\begin{array}{l}\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)=\frac{1}{2}\left(4\right)+3\\ =2+3\\ =5\end{array}$

Hence the limit of the function $\underset{x\to 4}{\lim }\left(\frac{1}{2}x+3\right)$ is $5$ .