Chapter 12, Problem 8RE

To determine

Evaluating a Limit by Direct Substitution

Answer to Problem 8RE

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

Explanation of Solution

Given information: find the limit by direct substitution.

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

Calculation:

The limit of given function is,

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

Now find the limit of the function by using direct substitution method, by replacing the value of $x$ by 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 property of limit $\underset{x\to c}{\lim }b=b$

  $\underset{x\to c}{\lim }\left[f\left(x\right)\pm g\left(x\right)\right]$

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

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

Replace $x$ by $4$

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

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