Chapter 2.1, Problem 12E

To determine

To find: The value of $\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x$ with the help of substitution. Check the results graphically.

Answer to Problem 12E

The value of $\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x$ is $0$ .

Explanation of Solution

Given information:

The given limit is $\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x$ .

Calculation:

Substitute $\frac{1}{2}$ for $x$ in the given limit.

  $\begin{array}{c}\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x=\mathrm{int}\frac{1}{2}\\ =0\end{array}$

Therefore, the value of $\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x$ is $0$ .

Check the limit graphically.

Follow the steps using graphing calculator to graph a function $f\left(x\right)=\mathrm{int}x$ .

First press “ON” button on graphical calculator. Press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=\mathrm{int}x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{MATH}}\boxed{\to }\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\mathrm{int}\left(\text{X}\right)$

Press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function. Enter the keystrokes for the value of the function at particular point $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{1}\boxed{1}\boxed{÷}\boxed{2}$ and press the $\boxed{\text{ENTER}}$ key.

The graph plotted in standard window as shown in Figure (1).

  

Figure (1)

It is verified graphically that the value of $\underset{x\to \frac{1}{2}}{\lim }\mathrm{int}x$ is $0$ .