Chapter 2.1, Problem 44E

(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 44E

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 is:

  

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

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

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

(b)

To determine

To check: Whether the statement $\underset{x\to 2}{\lim }f\left(x\right)$ doesn’t exist is true or not.

Answer to Problem 44E

The statement $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist is false.

Explanation of Solution

Given information:

The graph of the function is:

  

As observed from the graph, the function $y=f\left(x\right)$ approaches to $1$ as $x$ approaches to $2$ from both the sides of $x=2$ .

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

So, the limit $\underset{x\to 2}{\lim }f\left(x\right)$ exists.

Therefore, the statement $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist is false.

(c)

To determine

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

Answer to Problem 44E

The statement $\underset{x\to 2}{\lim }f\left(x\right)=2$ is false.

Explanation of Solution

Given information:

The graph of the function is:

  

As calculated in part (b), the left hand limit and right hand limit is equal to $1$ .

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

So, the value of $\underset{x\to 2}{\lim }f\left(x\right)$ is not equal to $2$ .

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

(d)

To determine

To check: Whether the statement $\underset{x\to 1^{-}}{\lim }f\left(x\right)=2$ is true or not.

Answer to Problem 44E

Yes, the statement $\underset{x\to 1^{-}}{\lim }f\left(x\right)=2$ is true.

Explanation of Solution

Given information:

The graph of the function is:

  

As observed from the graph, the function $y=f\left(x\right)$ approaches to $2$ as $x$ approaches to $1$ from the left side.

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

Therefore, the statement $\underset{x\to 1^{-}}{\lim }f\left(x\right)=2$ is true.

(e)

To determine

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

Answer to Problem 44E

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 is:

  

As observed from the graph, the function $y=f\left(x\right)$ has value equal to $1$ for every value $x$ in $\left[1,2\right]$ . So,

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

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

(f)

To determine

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

Answer to Problem 44E

Yes, the statement $\underset{x\to 1}{\lim }f\left(x\right)$ does not exist is true.

Explanation of Solution

Given information:

The graph of the function is:

  

As calculated in part (d), the value $\underset{x\to 1^{-}}{\lim }f\left(x\right)$ is equal to $2$ .

As calculated in part (e), the value $\underset{x\to 1^{+}}{\lim }f\left(x\right)$ is equal to $1$ .

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

Therefore, the statement $\underset{x\to 1}{\lim }f\left(x\right)$ does not exist is true.

(g)

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 44E

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 is:

  

As observed from the graph, the function $y=f\left(x\right)$ approaches to $0$ as $x$ approaches to $0$ from both the sides.

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

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

(h)

To determine

To check: Whether the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(-1,1\right)$ is true or not.

Answer to Problem 44E

Yes, the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(-1,1\right)$ .

Explanation of Solution

Given information:

The graph of the function is:

  

As observed from the graph, the function $y=f\left(x\right)$ is a continuous graph in the interval $\left(-1,1\right)$ .

So, the value $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(-1,1\right)$ .

Therefore, the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(-1,1\right)$ is true.

(i)

To determine

To check: Whether the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(1,3\right)$ is true or not.

Answer to Problem 44E

Yes, the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(1,3\right)$ is true.

Explanation of Solution

Given information:

The graph of the function is:

  

As observed from the graph, the function $y=f\left(x\right)$ has the value $1$ in the interval $\left(1,3\right)$ .

So, the value $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(1,3\right)$ .

Therefore, the statement $\underset{x\to c}{\lim }f\left(x\right)$ exists at every $c$ in $\left(1,3\right)$ is true.