https://file.aiquickdraw.com/mathbot/file/1726272313656y4kia6eg.jpg

The problem asks to find the \( x \) values for which the function \( f(x) \) is not continuous. The function is defined as a piecewise function:

\[
f(x) = 
\begin{cases} 
\frac{1}{x - 3} & \text{for } x \geq 5 \\
\frac{1}{2} & \text{for } x < 5
\end{cases}
\]

### Step-by-step solution:
1. **Continuity at \( x = 5 \)**:
   - To check if \( f(x) \) is continuous at \( x = 5 \), we need to verify if:
     - The limit from the left-hand side (LHS) and the right-hand side (RHS) as \( x \to 5 \) exists and are equal.
     - The value of \( f(5) \) matches the limit at \( x = 5 \).

   - **Right-hand limit** (\( x \to 5^+ \)): 
     For \( x \geq 5 \), \( f(x) = \frac{1}{x - 3} \).
     \[
     \lim_{x \to 5^+} f(x) = \frac{1}{5 - 3} = \frac{1}{2}
     \]

   - **Left-hand limit** (\( x \to 5^- \)): 
     For \( x < 5 \), \( f(x) = \frac{1}{2} \), so:
     \[
     \lim_{x \to 5^-} f(x) = \frac{1}{2}
     \]

   - Since both the left-hand and right-hand limits are equal and the value of \( f(5) = \frac{1}{2} \), the function is continuous at \( x = 5 \).

2. **Continuity at \( x = 3 \)**:
   - For \( x \geq 5 \), the expression \( \frac{1}{x - 3} \) is defined, but if \( x = 3 \), the denominator would become zero, making \( f(x) \) undefined. Hence, the function has a discontinuity at \( x = 3 \).

### Conclusion:
The function is not continuous at \( x = 3 \), but it is continuous at \( x = 5 \). Therefore, the correct answer is **(C) 3**.

Would you like more details or have any questions?

Here are five related questions:
1. How do we determine if a piecewise function is continuous at a point of transition?
2. What happens if a function has a discontinuity at an undefined point?
3. What are the different types of discontinuities (removable, jump, infinite)?
4. How do limits play a role in determining the continuity of a function?
5. What are the conditions for continuity of a function at a specific point?

**Tip:** When analyzing a piecewise function, always check the boundaries of the intervals where the pieces change. These are potential points of discontinuity.

Find the x values for which f(x) is not continuous if f(x) = 1/(x-3) for x >= 5 and 1/2 for x < 5.

Learn how to determine the points of discontinuity for a piecewise function where f(x) = 1/(x - 3) for x >= 5 and f(x) = 1/2 for x < 5. This involves understanding the concept of limits and continuity at a point, particularly at x = 3 and x = 5.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation