f(x)=(x-3)/(|x-3|) when x doesn't equal 3, 3 when x equals 3, which of the following statements are true? A. limit as x approaches 3 from the right f(x)=1 B. Limit as x approaches 3 from the left f(x)=-1 3. Limit as x approaches 3 f(x)=3

Let's evaluate each statement based on the function provided:

### Given function:
\[ f(x) = \frac{x-3}{|x-3|}, \text{ for } x \neq 3, \]
\[ f(3) = 3. \]

### 1. Evaluating \( \lim_{x \to 3^+} f(x) \):
- For \( x > 3 \), \( x - 3 > 0 \), so \( |x - 3| = x - 3 \).
- Therefore, when \( x > 3 \),
  \[
  f(x) = \frac{x - 3}{x - 3} = 1.
  \]
- Hence, \( \lim_{x \to 3^+} f(x) = 1 \).

### 2. Evaluating \( \lim_{x \to 3^-} f(x) \):
- For \( x < 3 \), \( x - 3 < 0 \), so \( |x - 3| = -(x - 3) \).
- Therefore, when \( x < 3 \),
  \[
  f(x) = \frac{x - 3}{-(x - 3)} = -1.
  \]
- Hence, \( \lim_{x \to 3^-} f(x) = -1 \).

### 3. Evaluating \( \lim_{x \to 3} f(x) \):
- Since the left-hand limit is \( -1 \) and the right-hand limit is \( 1 \), the overall limit as \( x \to 3 \) does **not** exist.
- Thus, \( \lim_{x \to 3} f(x) \neq 3 \) and in fact, doesn't exist at all.

### Conclusion:
- **Statement A (Limit as x approaches 3 from the right \( f(x) = 1 \))** is **true**.
- **Statement B (Limit as x approaches 3 from the left \( f(x) = -1 \))** is **true**.
- **Statement C (Limit as x approaches 3 \( f(x) = 3 \))** is **false**.

Would you like any more details or have further questions?

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Here are 5 related questions you might find useful:
1. What is the definition of a one-sided limit?
2. How do we check if a function is continuous at a point?
3. How do absolute value functions affect limits?
4. What are the conditions for a limit to exist at a point?
5. Can a limit exist if the left-hand and right-hand limits are different?

**Tip:** Always check the left-hand and right-hand limits separately when dealing with absolute value functions or piecewise functions.

f(x)=(x-3)/(|x-3|) when x doesn't equal 3, 3 when x equals 3, which of the following statements are true? A. limit as x approaches 3 from the right f(x)=1 B. Limit as x approaches 3 from the left f(x)=-1 C. Limit as x approaches 3 f(x)=3

This page explores how to evaluate limits for the piecewise function f(x) = (x-3)/(|x-3|) when x ≠ 3 and f(3) = 3. Detailed explanations include checking one-sided limits and understanding the role of absolute values in limit behavior. Suitable for high school Pre-Calculus and Calculus students.

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