Chapter 11.3, Problem 82E

Solution

a)

To draw: The graph of $f\left(x\right)=\left\{\begin{array}{l}{x}^2\text{ if -2}\le x<0\text{ or 0}<x\le \text{2}\\ \text{1 if }x=0\\ \text{2}x\text{ if }x<-2\text{ or}x>\text{2}\end{array}\right.$ .

For the given piecewise function the first piece is $y=x^2$ with vertex at (0,0) and second piece is (0,1) and third piece is $y=2x$ which is a line.

Given information: The given piecewise defined function is $f\left(x\right)=\left\{\begin{array}{l}{x}^2\text{ if -2}\le x<0\text{ or 0}<x\le \text{2}\\ \text{1 if }x=0\\ \text{2}x\text{ if }x<-2\text{ or}x>\text{2}\end{array}\right.$

Calculation:

For the given piecewise function, the first piece is $y=x^2$ with vertex at origin which is a parabola that opens upward and second piece is point (0,1) and third piece is a line $y=2x$ with slope 2 and y -intercept at (0,0).

Thus, the graph is plotted below:

  

b)

To find: The points $c$ in the domain of $f$where $\underset{x\to c}{\lim }f\left(x\right)$ exists.

The limit $\underset{x\to c}{\lim }f\left(x\right)$ exists and this occurs for all $x\in \left(-\infty ,-2\right)\cup \left(-2,\infty \right)$ .

Given information: The given piecewise defined function is $f\left(x\right)=\left\{\begin{array}{l}{x}^2\text{ if -2}\le x<0\text{ or 0}<x\le \text{2}\\ \text{1 if }x=0\\ \text{2}x\text{ if }x<-2\text{ or}x>\text{2}\end{array}\right.$

Calculation:

The limit $\underset{x\to c}{\lim }f\left(x\right)$ exists if and is equal to $L$ if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=L$ and this occurs for all $x\in \left(-\infty ,-2\right)\cup \left(-2,\infty \right)$ .

c)

To find: The points $c$ where only the left-hand limit exists.

The left-hand limit of $\underset{x\to c}{\lim }f\left(x\right)$ at point $c$ exists at none.

Given information: The given piecewise defined function is $f\left(x\right)=\left\{\begin{array}{l}{x}^2\text{ if -2}\le x<0\text{ or 0}<x\le \text{2}\\ \text{1 if }x=0\\ \text{2}x\text{ if }x<-2\text{ or}x>\text{2}\end{array}\right.$

Calculation:

The left-hand limit at point $c$ exists when there is no graph to right of $x=c$ and when right hand limit is $\infty /-\infty$ .

Thus, the left-hand limit of $\underset{x\to c}{\lim }f\left(x\right)$ at point $c$ exists at none.

d)

To find: The points $c$ where only the right-hand limit exists.

The right-hand limit of $\underset{x\to c}{\lim }f\left(x\right)$ at point $c$ exists at none.

Given information: The given piecewise defined function is $f\left(x\right)=\left\{\begin{array}{l}{x}^2\text{ if -2}\le x<0\text{ or 0}<x\le \text{2}\\ \text{1 if }x=0\\ \text{2}x\text{ if }x<-2\text{ or}x>\text{2}\end{array}\right.$

Calculation:

The right-hand limit at point $c$ exists when there is no graph to left of $x=c$ and when left hand limit is $\infty /-\infty$ .

Thus, the right-hand limit of $\underset{x\to c}{\lim }f\left(x\right)$ at point $c$ exists at none.