Chapter 2.1, Problem 43E

(a)

To determine

To check: Whether the statement $\underset{x\to -1^{+}}{\lim }f\left(x\right)=1$ is true or not.

Answer to Problem 43E

Yes, the statement $\underset{x\to -1^{+}}{\lim }f\left(x\right)=1$ is true.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in the graph, the function $y=f\left(x\right)$ approaches to $1$ as $x$ approaches to $-1$ from the right side.

So, the value of $\underset{x\to -1^{+}}{\lim }f\left(x\right)$ is equal to $1$ .

Therefore, the statement $\underset{x\to -1^{+}}{\lim }f\left(x\right)=1$ is true.

(b)

To determine

To check: Whether the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=0$ is true or not.

Answer to Problem 43E

Yes, the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=0$ is true.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in the graph, the function $y=f\left(x\right)$ approaches to $0$ as $x$ approaches to $0$ from the left side.

So, the value of $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ is equal to $0$ .

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

(c)

To determine

To check: Whether the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=1$ is true or not.

Answer to Problem 43E

No, the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=1$ is false.

Explanation of Solution

Given information:

The graph of the function:

  

From the graph it can be observed that, the function $y=f\left(x\right)$ approaches to $0$ as $x$ approaches to $0$ from the left side. So, the value of $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ is not equal to $1$ .

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

(d)

To determine

To check: Whether the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)$ is true or not.

Answer to Problem 43E

Yes, the statement $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)$ is true.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in the graph, the function $y=f\left(x\right)$ approaches to $0$ as $x$ approaches to $-1$ from both the sides of $x=0$ . So, the value of $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ and $\underset{x\to 0^{+}}{\lim }f\left(x\right)$ is equal to $0$ .

  $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)$

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

(e)

To determine

To check: Whether the statement $\underset{x\to 0}{\lim }f\left(x\right)$ exists is true or not.

Answer to Problem 43E

Yes, the statement $\underset{x\to 0}{\lim }f\left(x\right)$ exists is true.

Explanation of Solution

Given information:

The graph of the function:

  

As calculated in part (d), the value of both $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ and $\underset{x\to 0^{+}}{\lim }f\left(x\right)$ is equal to $0$ . So, the value of $\underset{x\to 0}{\lim }f\left(x\right)$ exists.

Therefore, the statement $\underset{x\to 0}{\lim }f\left(x\right)$ exists is true.

(f)

To determine

To check: Whether the statement $\underset{x\to 0}{\lim }f\left(x\right)=0$ is true or not.

Answer to Problem 43E

Yes, the statement $\underset{x\to 0}{\lim }f\left(x\right)=0$ is true.

Explanation of Solution

Given information:

The graph of the function:

  

As calculated in part (d), the value of both $\underset{x\to 0^{-}}{\lim }f\left(x\right)$ and $\underset{x\to 0^{+}}{\lim }f\left(x\right)$ is equal to $0$ . So,

  $\underset{x\to 0}{\lim }f\left(x\right)=0$

Therefore, the statement $\underset{x\to 0}{\lim }f\left(x\right)=0$ is true.

(g)

To determine

To check: Whether the statement $\underset{x\to 0}{\lim }f\left(x\right)=1$ is true or not.

Answer to Problem 43E

No, the statement $\underset{x\to 0}{\lim }f\left(x\right)=1$ is false.

Explanation of Solution

Given information:

The graph of the function:

  

As calculated in part (f), the value of $\underset{x\to 0}{\lim }f\left(x\right)$ is equal to $0$ . So, the value of $\underset{x\to 0}{\lim }f\left(x\right)$ is not equal to $1$ .

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

(h)

To determine

To check: Whether the statement $\underset{x\to 1}{\lim }f\left(x\right)=1$ is true or not.

Answer to Problem 43E

No, the statement $\underset{x\to 1}{\lim }f\left(x\right)=1$ is false.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in the graph, the function $y=f\left(x\right)$ approaches to $1$ as $x$ approaches to $1$ from the left side. So,

  $\underset{x\to 1^{-}}{\lim }f\left(x\right)=1$

Also it can be observed from the graph that the function $y=f\left(x\right)$ has value equal to $0$ in the interval $\left[1,2\right]$ . So,

  $\underset{x\to 1^{+}}{\lim }f\left(x\right)=0$

Both the limits are not equal. So, the $\underset{x\to 1}{\lim }f\left(x\right)$ does not exist.

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

(i)

To determine

To check: Whether the statement $\underset{x\to 1}{\lim }f\left(x\right)=0$ is true or not.

Answer to Problem 43E

No, the statement $\underset{x\to 1}{\lim }f\left(x\right)=0$ is false.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in part (h), the limit $\underset{x\to 1}{\lim }f\left(x\right)$ does not exist. So, the value of $\underset{x\to 1}{\lim }f\left(x\right)$ is not equal to $0$ .

Therefore, the statement $\underset{x\to 1}{\lim }f\left(x\right)$ is false.

(j)

To determine

To check: Whether the statement $\underset{x\to 2^{-}}{\lim }f\left(x\right)=2$ is true or not.

Answer to Problem 43E

No, the statement $\underset{x\to 2^{-}}{\lim }f\left(x\right)=2$ is false.

Explanation of Solution

Given information:

The graph of the function:

  

As shown in the graph, the function $y=f\left(x\right)$ has value equal to $0$ in the interval $\left[1,2\right]$ . So,

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=0$

Therefore, the statement $\underset{x\to 2^{-}}{\lim }f\left(x\right)=2$ is false.