Chapter 12, Problem 3PS

a.

To determine

Evaluate, $f\left(\frac{1}{4}\right)$ , $f(3)$ and $f(1)$

Answer to Problem 3PS

  $\boxed{4}$ , $\boxed{0}$ and $\boxed{1}$

Explanation of Solution

Given information:

Sketch the graph of the function $f(x)=〚\frac{1}{2}〛$ .

Evaluate, $f\left(\frac{1}{4}\right)$ , $f(3)$ and $f(1)$ .

Calculation:

Consider the following limit

  $f(x)=〚\frac{1}{2}〛$

To sketch the graph of the function, proceed as follows.

Press $Y$ and provide the function. The display will be

  

Now press window and choose proper scale. The display will be

  

Now press graph, the display will be

  

Now to evaluate the value of the function $f(x)=〚\frac{1}{2}〛$ , use the definition of maximum integer function.

Definition of maximum integre function:

  $\begin{array}{l}[1.05]=1\\ [-1.05]=-2\end{array}$

Therefore

The value of function $f(x)=〚\frac{1}{2}〛$ at $x=\frac{1}{4}$

  $\begin{array}{l}f\left(\frac{1}{4}\right)=\left[4\right]\\ =\boxed{4}\end{array}$

The value of function $f(x)=〚\frac{1}{2}〛$ at $x=3$

  $\begin{array}{l}f(x)=\left[\frac{1}{3}\right]\\ =\left[0.33\right]\\ =\boxed{0}\end{array}$

The value of function $f(x)=〚\frac{1}{2}〛$ at $x=1$

  $\begin{array}{l}f(1)=[1]\\ =[1]\\ =\boxed{1}\end{array}$

b.

To determine

Evaluate the following limits.

  $\underset{x\to 1^{-}}{\lim }f(x),\underset{x\to 1^{+}}{\lim }f(x),\underset{x\to {\left(1/2\right)}^{-}}{\lim }f(x),\underset{x\to {\left(1/2\right)}^{+}}{\lim }$ .

Answer to Problem 3PS

  $\boxed{1}$ , $\boxed{0}$ , $\boxed{2}$ and $\boxed{1}$

Explanation of Solution

Given information:

Sketch the graph of the function $f(x)=〚\frac{1}{2}〛$ .

Evaluate the following limits.

  $\underset{x\to 1^{-}}{\lim }f(x),\underset{x\to 1^{+}}{\lim }f(x),\underset{x\to {\left(1/2\right)}^{-}}{\lim }f(x),\underset{x\to {\left(1/2\right)}^{+}}{\lim }$

Calculation:

Consider the following limit

  $f(x)=〚\frac{1}{2}〛$

To sketch the graph of the function, proceed as follows.

Press $Y$ and provide the function. The display will be

  

Now press window and choose proper scale. The display will be

  

Now press graph, the display will be

  

To evaluate the function $f(x)=〚\frac{1}{2}〛$ at different limit as follows.

  $\begin{array}{l}\underset{x\to 1^{-}}{\lim }f(x)=\underset{h\to 0}{\lim }\left[\frac{1}{1-h}\right]\\ =\left[>1\right]\\ =\boxed{1}\end{array}$

  $\begin{array}{l}\underset{x\to 1^{+}}{\lim }f(x)=\underset{h\to 0}{\lim }\left[\frac{1}{1+h}\right]\\ =\left[<1\right]\\ =\boxed{0}\end{array}$

Further evaluate

  $\begin{array}{l}\underset{x\to {(1/2)}^{-}}{\lim }f(x)=\underset{h\to 0}{\lim }\left[\frac{1}{\frac{1}{2}-h}\right]\\ =\underset{h\to 0}{\lim }\left[\frac{2}{1-2h}\right]\\ =\boxed{2}\end{array}$

  $\begin{array}{l}\underset{x\to {(1/2)}^{+}}{\lim }f(x)=\underset{h\to 0}{\lim }\left[\frac{1}{\frac{1}{2}+h}\right]\\ =\underset{h\to 0}{\lim }\left[\frac{2}{1+2h}\right]\\ =\boxed{1}\end{array}$