https://file.aiquickdraw.com/mathbot/file/1727062338399z38w7phe.jpg

Let's analyze the given function \( f(x) \):

\[
f(x) = 
\begin{cases} 
\frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\
1 & \text{if } x = 2 
\end{cases}
\]

We need to determine which statements about \( f \) are true:

### I. \( f \) has a limit at \( x = 2 \)

To check the limit as \( x \) approaches 2, we simplify the expression for \( f(x) \) when \( x \neq 2 \).

\[
f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for } x \neq 2.
\]

Thus, for \( x \neq 2 \), \( f(x) = x + 2 \).

Now, we check the limit as \( x \) approaches 2:

\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 4.
\]

Since \( f(2) = 1 \), the value of the function at \( x = 2 \) is not equal to the limit as \( x \) approaches 2. However, **the limit exists** and equals 4, so statement I is **true**.

### II. \( f \) is continuous at \( x = 2 \)

For continuity at \( x = 2 \), the following must hold:
1. \( \lim_{x \to 2} f(x) \) exists (which we found to be 4),
2. \( f(2) \) exists (we know \( f(2) = 1 \)),
3. \( \lim_{x \to 2} f(x) = f(2) \).

Since \( \lim_{x \to 2} f(x) = 4 \) and \( f(2) = 1 \), the function is **not continuous** at \( x = 2 \). Therefore, statement II is **false**.

### III. \( f \) is differentiable at \( x = 2 \)

For differentiability, the function must be continuous at \( x = 2 \). Since we just established that \( f \) is not continuous at \( x = 2 \), it cannot be differentiable at that point. Thus, statement III is **false**.

### Conclusion:
The only true statement is I, so the correct answer is **A) I only**.

---

Would you like more details on any part of the solution?

Here are five related questions you might find helpful:
1. What is the definition of continuity, and how is it related to limits?
2. Can a function have a limit but still be discontinuous at a point?
3. What is the formal definition of differentiability, and how does it relate to continuity?
4. Can a function be differentiable but not continuous? Why or why not?
5. How do you simplify rational functions like \( \frac{x^2 - 4}{x - 2} \) for limits?

**Tip**: Always check for removable discontinuities by simplifying the function before finding limits or continuity.

Let f be the function defined as f(x) = (x^2 - 4) / (x - 2) if x ≠ 2 and f(x) = 1 if x = 2. Which of the following statements are true? I. f has a limit at x = 2, II. f is continuous at x = 2, III. f is differentiable at x = 2.

Explore the concepts of limits, continuity, and differentiability using a piecewise function defined as f(x) = (x^2 - 4) / (x - 2) for x ≠ 2 and f(x) = 1 for x = 2. Solve for the limit at x = 2, check continuity, and determine if the function is differentiable.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation