Chapter 11.3, Problem 31E

Solution

a.

To find whether the statement $\underset{x\to -1^{+}}{\lim }f(x)=1$ is true or false for the given function $f(x)$ .

True

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches $1$ from the right of the point $x=-1$ . That means, the limit of the function at $x=-1$ from right is 1.

That is, $\underset{x\to -1^{+}}{\lim }f(x)=1$ .

Therefore, the statement is true.

Conclusion:

The statement is true.

b.

To find whether the statement $\underset{x\to 0^{-}}{\lim }f(x)=0$ is true or false for the given function $f(x)$ .

True

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

Therefore, the statement is true.

Conclusion:

The statement is true.

c.

To find whether the statement $\underset{x\to 0^{-}}{\lim }f(x)=1$ is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

Therefore, the statement $\underset{x\to 0^{-}}{\lim }f(x)=1$ is false.

Conclusion:

The statement is false.

d.

To find whether the statement $\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }f(x)$ is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

The function approaches 1 from the right of the point $x=0$ . Thus, $\underset{x\to 0^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 0^{-}}{\lim }f(x)\ne \underset{x\to 0^{+}}{\lim }f(x)$ .

Therefore, the statement $\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }f(x)$ is false.

Conclusion:

The statement is false.

e.

To find whether the statement “ $\underset{x\to 0}{\lim }f(x)$ exists” is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

The function approaches 1 from the right of the point $x=0$ . Thus, $\underset{x\to 0^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 0^{-}}{\lim }f(x)\ne \underset{x\to 0^{+}}{\lim }f(x)$ .

Hence, $\underset{x\to 0}{\lim }f(x)$ does not exist.

Conclusion:

The statement is false.

f.

To find whether the statement “ $\underset{x\to 0}{\lim }f(x)=0$ ” is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

The function approaches 1 from the right of the point $x=0$ . Thus, $\underset{x\to 0^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 0^{-}}{\lim }f(x)\ne \underset{x\to 0^{+}}{\lim }f(x)$ .

Hence, $\underset{x\to 0}{\lim }f(x)$ does not exist.

Conclusion:

The statement is false.

g.

To find whether the statement “ $\underset{x\to 0}{\lim }f(x)=1$ ” is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 0 from the left of the point $x=0$ . That means, the limit of the function at $x=0$ from left is 0.

That is, $\underset{x\to 0^{-}}{\lim }f(x)=0$ .

The function approaches 1 from the right of the point $x=0$ . Thus, $\underset{x\to 0^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 0^{-}}{\lim }f(x)\ne \underset{x\to 0^{+}}{\lim }f(x)$ .

Hence, $\underset{x\to 0}{\lim }f(x)$ does not exist.

Conclusion:

The statement is false.

h.

To find whether the statement “ $\underset{x\to 1}{\lim }f(x)=1$ ” is true or false for the given function $f(x)$ .

True

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 1 from the left of the point $x=1$ . That means, the limit of the function at $x=1$ from left is 1.

That is, $\underset{x\to 1^{-}}{\lim }f(x)=1$ .

The function approaches 1 from the right of the point $x=1$ . Thus, $\underset{x\to 1^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }f(x)$ .

Hence, $\underset{x\to 1}{\lim }f(x)=\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }f(x)=1$ .

Conclusion:

The statement is true.

i.

To find whether the statement “ $\underset{x\to 1}{\lim }f(x)=0$ ” is true or false for the given function $f(x)$ .

False

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 1 from the left of the point $x=1$ . That means, the limit of the function at $x=1$ from left is 1.

That is, $\underset{x\to 1^{-}}{\lim }f(x)=1$ .

The function approaches 1 from the right of the point $x=1$ . Thus, $\underset{x\to 1^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }f(x)$ .

Hence, $\underset{x\to 1}{\lim }f(x)=\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }f(x)=1$ .

Therefore, $\underset{x\to 1}{\lim }f(x)\ne 0$ .

Conclusion:

The statement is false.

j.

To find whether the statement “ $\underset{x\to 2^{-}}{\lim }f(x)=2$ ” is true or false for the given function $f(x)$ .

True

Given information:

The graph of the function $y=f\left(x\right)$ .

  

Calculation:

Observe that the function approaches 2 from the left of the point $x=2$ . That means, the limit of the function at $x=2$ from left is 2.

That is, $\underset{x\to 2^{-}}{\lim }f(x)=2$ .

Conclusion:

The statement is true.