Determine whether the following statements are true and give an explanation or counterexample. Complete parts​ (a) through​ (e).
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Part 1
a. The value of ModifyingBelow lim With x right arrow 4 StartFraction x squared minus 16 Over x minus 4 EndFractionlimx→4
x2−16
x−4 does not exist. Choose the correct answer below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice.
A.
This statement is false because ModifyingBelow lim With x right arrow 4 StartFraction x squared minus 16 Over x minus 4 EndFractionlimx→4
x2−16
x−4equals=enter your response here. 
​(Simplify your​ answer.)
B.
This statement is true because both corresponding​ one-sided limits exist but ModifyingBelow lim With x right arrow 4 Superscript minus StartFraction x squared minus 16 Over x minus 4 EndFractionlimx→4−
x2−16
x−4equals=enter your response here and ModifyingBelow lim With x right arrow 4 Superscript plus StartFraction x squared minus 16 Over x minus 4 EndFractionlimx→4+
x2−16
x−4equals=enter your response here. 
​(Simplify your​ answers.)
C.
This statement is true because 44 is not in the domain of ​f(x)equals=StartFraction x squared minus 16 Over x minus 4 EndFraction
x2−16
x−4.
D.
This statement is true because at least one of the corresponding​ one-sided limits does not exist.

Let's work through the given limit problem.

The function in question is:

\[
f(x) = \frac{x^2 - 16}{x - 4}
\]

We are asked to evaluate the limit of this function as \( x \to 4 \), specifically:

\[
\lim_{x \to 4} \frac{x^2 - 16}{x - 4}
\]

### Step 1: Simplify the expression

Notice that the numerator can be factored as a difference of squares:

\[
x^2 - 16 = (x - 4)(x + 4)
\]

Thus, the function becomes:

\[
f(x) = \frac{(x - 4)(x + 4)}{x - 4}
\]

For all \( x \neq 4 \), we can cancel out the common factor \( x - 4 \):

\[
f(x) = x + 4
\]

### Step 2: Evaluate the limit

The function now simplifies to \( f(x) = x + 4 \), which is a continuous function. Thus, we can directly substitute \( x = 4 \) into this simplified expression:

\[
\lim_{x \to 4} (x + 4) = 4 + 4 = 8
\]

### Step 3: Analyze the options

- **Option A:** This statement is **false** because the limit does exist, and it equals \( 8 \). Therefore, the correct value is:

  \[
  \lim_{x \to 4} \frac{x^2 - 16}{x - 4} = 8
  \]

- **Option B:** This statement is **false** because the one-sided limits exist and are equal. Both the left-hand limit \( \lim_{x \to 4^-} \) and the right-hand limit \( \lim_{x \to 4^+} \) are \( 8 \).

- **Option C:** This statement is **false** because \( x = 4 \) is a removable discontinuity. We can define \( f(4) = 8 \), and the limit exists.

- **Option D:** This statement is **false** because both one-sided limits exist, and they are equal to \( 8 \).

### Conclusion

The correct choice is **Option A**, and the limit is:

\[
\lim_{x \to 4} \frac{x^2 - 16}{x - 4} = 8
\]

Would you like further details or clarifications?

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### Related Questions:
1. What is a removable discontinuity, and how does it apply here?
2. How can we use factoring to simplify other types of limit problems?
3. What are the conditions under which a limit does not exist?
4. Can L'Hôpital's Rule be used to solve this problem? Why or why not?
5. How do we evaluate limits involving indeterminate forms?

**Tip:** Always check for ways to simplify algebraic expressions before applying more advanced techniques like L'Hôpital's Rule.

Determine whether the following statements are true and give an explanation or counterexample. Complete parts​ (a) through​ (e). Part 1 a. The value of limx→4 (x² - 16) / (x - 4) does not exist. Choose the correct answer below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice. A. This statement is false because limx→4 (x² - 16) / (x - 4) equals 8. B. This statement is true because both corresponding​ one-sided limits exist but limx→4⁻ (x² - 16) / (x - 4) equals __ and limx→4⁺ (x² - 16) / (x - 4) equals __. C. This statement is true because 4 is not in the domain of f(x) = (x² - 16) / (x - 4). D. This statement is true because at least one of the corresponding​ one-sided limits does not exist.

This detailed solution explains how to evaluate the limit of (x² - 16) / (x - 4) as x approaches 4, covering key concepts like factoring, removable discontinuities, and continuous functions. Suitable for students in Grades 11-12 studying limits in calculus.

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