Chapter 1.1, Problem 44E

(a)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to 1^{+}}{\lim }f(x)=1$

Calculation:

The above statement is understood to be the limit of $f(x)$ as $x$ approaches -1 from the right; as a result, one must trace the function's graph from the right and go toward $x=-1$ . This value approaches 1, so the assertion is accurate, as you can see when you do that and look at the associated function value on the y-axis.

Therefore the statement is true.

(b)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is false.

Explanation of Solution

Given information: $\underset{x\to 2}{\lim }f(x)$ does not exists.

Calculation:

The given statement might be interpreted as meaning that there is no $f(x)$ limit as x approaches 2. Must locate both the right-hand and left-hand limits to determine whether they are equal and whether they exist in order to determine whether this is true.

Follow the function's graph from the left and get close to $x=2$ to get the left-hand limit. see that this number is approaching zero when look at the matching function value on the y-axis. 1

Follow the function's graph from the right and get close to $x=2$ to get the right-hand limit. This value approaches 1 when look at the associated function value on the y-axis.

The left-hand and right-hand limits both exist and are equal at 1, therefore the limit does exist and the assertion is false.

(c)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is false.

Explanation of Solution

Given information: $\underset{x\to 2}{\lim }f(x)=2$

Calculation:

As x gets closer to 2, the limit of $f(x)$ is read as the number 2. determine the right-hand and left-hand limits and check to see if they are equal to 2 to determine if this is true.

Follow the function's graph from the left and get close to $x=2$ to get the left-hand limit. see that this number is getting close to 1 when read the matching function value on the y-axis.

Follow the function's graph from the right and get close to $x=2$ to get the right-hand limit. This value approaches 1 when look at the associated function value on the y-axis.

The claim is false because the left-hand and right-hand limits are 1 and not 2, respectively.

(d)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to 1^{-}}{\lim }f(x)=2$

Calculation:

According to the given assertion, the limit of $f(x)$ as x moves toward 1 from the left is 2. Therefore, one must follow the function's graph as it moves toward $x=1$ from the left. The assertion is true because you can see that the value of the corresponding function on the y-axis approaches 2.

Therefore the required given statement is true.

(e)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to 1^{+}}{\lim }f(x)=1$

Calculation:

If trace the graph as move toward $x=1$ from the right, we discover that the graph moves toward point $(1,1)$ , indicating that the limit is 1. The assertion is accurate.

Therefore the required given statement is true.

(f)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to 1}{\lim }f(x)$ does not exists

Calculation:

If the left and right have different boundaries, then there won't be a limit. In fact, discovered:

  $\begin{array}{c}\underset{x\to 1^{-}}{\lim }f(x)=2\\ \underset{x\to 1^{+}}{\lim }f(x)=1\end{array}$

The limit as defined by $x\to 1$ does not exist because the bounds are distinct. The claim is true.

Therefore the required given statement is true.

(g)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to 0^{+}}{\lim }f(x)=\underset{x\to 0^{-}}{\lim }f(x)$

Calculation:

If the limit on both sides is the same as x gets closer to zero. The graph reaches the point $(0,0)$ if we follow it from left to right towards $x=0$ .

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

Now if follow the graph's path from the right, see that it tends to head in the same direction.

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

The statement is true because the boundaries are equal.

(h)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to c}{\lim }f(x)$ exists at every $c$ in $(-1,1)$

Calculation:

There is a limit for every c in the open interval $(-1,1)$ since there are no vertical asymptotes or jump discontinuities there. The claim is true.

(i)

To determine

To check: The statement is true or false.

Answer to Problem 44E

The statement is true.

Explanation of Solution

Given information: $\underset{x\to c}{\lim }f(x)$ exists at every $c$ in $(1,3)$

Calculation:

This is accurate since the interval notation $(1,3)$ designates any x values between 1 and 3 that are not 1 or 3. write it out as $1<x<3$ instead. As long as there isn't a vertical asymptote or a jump discontinuity, the limit will be present. The limit will thus exist for all values between 1 and 3, as neither of these occur between 1 and 3.

Therefore the required given statement is true.