Chapter 2.1, Problem 63E

(a)

To determine

To graph: The piecewise function $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ .

Graph:

To graph $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key. To write the function $f\left(x\right)=\sqrt{1-x^2}$ for the interval $0\le x<1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes as given below:

  $\boxed{2\text{nd}}\boxed{x^2}\boxed{1}\boxed{-}\boxed{\text{X}}\boxed{x^2}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{0}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{1}\boxed{)}$

Press Enter. To write the function $f\left(x\right)=1$ for the interval $1\le x<2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes as given below:

  $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{1}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{2}\boxed{)}$

Again press Enter. To write the function $f\left(x\right)=2$ for $x=2$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes as given below:

  $\boxed{2}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{1}\boxed{2}\boxed{)}$

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\sqrt{\left(1-{\text{X}}^2\right)}/\left(\text{X}\ge 0\text{and X}<1\right)\\ {\text{Y}}_2=1/\left(\text{X}\ge 1\text{and}\text{X}<2\right)\\ {\text{Y}}_3=2/\left(\text{X}=2\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

Interpretation: The graph of the piecewise function shows a quarter part of a circle in Quadrant I, horizontal line and a point.

(b)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that $\underset{x\to c}{\lim }f\left(x\right)$ exist for the given function.

Answer to Problem 63E

The limit $\underset{x\to c}{\lim }f\left(x\right)$ exist for every $c$ in the interval $\left(0,1\right)\cup \left(1,2\right)$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ .

Calculation:

The function $f\left(x\right)=\sqrt{1-x^2}$ corresponds for all the values of $x$ in $\left(0,1\right)$ . So, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exists for every $x$ in the interval $\left(0,1\right)$ .

The function exist for all points in the interval $\left(1,2\right)$ . So, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exists for every $x$ in the interval $\left(0,1\right)\cup \left(1,2\right)$ .

Therefore, the limit $\underset{x\to c}{\lim }f\left(x\right)$ exist for every $c$ in the interval $\left(0,1\right)\cup \left(1,2\right)$ .

(c)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that only left hand limit exist at $x=c$ .

Answer to Problem 63E

The point $c$ in the domain of $f\left(x\right)$ such that only left hand limit exists is 2.

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ .

Calculation:

As observed from the graph in part (a), the function is defined for $x=2$ from the left side but not from the right side of $x=2$ .

So, only left hand limit of function $f\left(x\right)$ exist for the point $x=2$ .

Therefore, the point $c$ in the domain of $f\left(x\right)$ such that only left hand limit exists is 2.

(d)

To determine

To find: The points $c$ in the domain of $f\left(x\right)$ such that only right hand limit exist at $x=c$ .

Answer to Problem 63E

The point $c$ in the domain of $f\left(x\right)$ such that only right hand limit exist is 0.

Explanation of Solution

Given information: The function is $f\left(x\right)=\{\begin{array}{l}\sqrt{1-x^2},0\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}$ .

Calculation:

As observed from the graph in part (a), the function is defined for $x=0$ from the right side but not the left side of $x=0$ .

So, only right hand limit of function $f\left(x\right)$ exist for the point $x=0$ .

Therefore, the point $c$ in the domain of $f\left(x\right)$ such that only right hand limit exist is 0.