Explain why the function is discontinuous at the given number a. (Select all that apply.)
f(x) = 

x2 − 4x
x2 − 16
    if x ≠ 4
1	    if x = 4
    
a = 4

f(4) is defined and 
lim 
x→4 
f(x) is finite, but they are not equal.

lim 
x→4+ 
f(x) and 
lim 
x→4− 
f(x) are finite, but are not equal.

f(4) is undefined.

lim 
x→4 
f(x) does not exist.
none of the above

Let's analyze the given function \( f(x) \) and determine why it might be discontinuous at \( a = 4 \).

The function is defined as:

\[
f(x) = \frac{x^2 - 4x}{x^2 - 16}, \text{ if } x \neq 4
\]
\[
f(4) = 1, \text{ if } x = 4
\]

### Step 1: Simplify the function for \( x \neq 4 \)

The denominator \( x^2 - 16 \) can be factored as:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

So the function becomes:
\[
f(x) = \frac{x(x - 4)}{(x - 4)(x + 4)}, \text{ for } x \neq 4
\]

For \( x \neq 4 \), we can cancel out the \( (x - 4) \) terms:
\[
f(x) = \frac{x}{x + 4}, \text{ for } x \neq 4
\]

### Step 2: Analyze the limits as \( x \to 4 \)

Now let's compute the limit of \( f(x) \) as \( x \to 4 \).

\[
\lim_{x \to 4} f(x) = \lim_{x \to 4} \frac{x}{x + 4} = \frac{4}{4 + 4} = \frac{4}{8} = \frac{1}{2}
\]

### Step 3: Compare \( f(4) \) and the limit as \( x \to 4 \)

- The function value at \( x = 4 \) is defined as \( f(4) = 1 \).
- The limit as \( x \to 4 \) is \( \frac{1}{2} \).

Since \( f(4) = 1 \) and \( \lim_{x \to 4} f(x) = \frac{1}{2} \), they are **not equal**. This discrepancy means the function is discontinuous at \( x = 4 \).

### Step 4: Analyze the options

1. **\( f(4) \) is defined and \( \lim_{x \to 4} f(x) \) is finite, but they are not equal.**  
   This statement is **true** because \( f(4) = 1 \) and \( \lim_{x \to 4} f(x) = \frac{1}{2} \), and they are not equal.

2. **\( \lim_{x \to 4^+} f(x) \) and \( \lim_{x \to 4^-} f(x) \) are finite, but are not equal.**  
   This statement is **false** because the limit from both sides is the same:  
   \( \lim_{x \to 4^+} f(x) = \lim_{x \to 4^-} f(x) = \frac{1}{2} \).

3. **\( f(4) \) is undefined.**  
   This statement is **false** because \( f(4) \) is explicitly defined as 1.

4. **\( \lim_{x \to 4} f(x) \) does not exist.**  
   This statement is **false** because the limit does exist and is \( \frac{1}{2} \).

### Conclusion

The correct answer is:
- \( f(4) \) is defined and \( \lim_{x \to 4} f(x) \) is finite, but they are not equal.

Would you like to dive deeper into this analysis or have further questions?

---

Here are 5 related questions to consider:

1. What is the formal definition of continuity at a point?
2. How would the function behave at points close to but not equal to 4?
3. Could \( f(x) \) be made continuous by redefining \( f(4) \)? How?
4. How does factoring the denominator help simplify functions with potential discontinuities?
5. What is the difference between removable and non-removable discontinuities?

**Tip:** Always check the function's definition and limits from both sides when analyzing continuity at a specific point.

Explain why the function is discontinuous at the given number a. (Select all that apply.) f(x) = (x^2 − 4x) / (x^2 − 16) if x ≠ 4, 1 if x = 4, a = 4.

Analyze the discontinuity of the function f(x) = (x^2 − 4x) / (x^2 − 16) at a = 4. Learn why f(4) is defined, but the limit does not match the function's value at this point. This explanation covers key concepts of limits, continuity, and rational functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation