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The problem asks you to find the indicated one-sided limits of the piecewise function \( f(x) \) and determine the continuity of \( f \) at the indicated point, \( x = 5 \). The function \( f(x) \) is given as:

\[
f(x) =
\begin{cases}
4 + 2x, & x < 5 \\
13 - 2x, & x \geq 5
\end{cases}
\]

Let's solve it step-by-step:

### 1. Finding the Left-Hand Limit \( \lim_{x \to 5^-} f(x) \)
- For \( x < 5 \), the function is \( f(x) = 4 + 2x \).
- The left-hand limit as \( x \to 5^- \) is:
  \[
  \lim_{x \to 5^-} f(x) = 4 + 2(5) = 4 + 10 = 14
  \]

### 2. Finding the Right-Hand Limit \( \lim_{x \to 5^+} f(x) \)
- For \( x \geq 5 \), the function is \( f(x) = 13 - 2x \).
- The right-hand limit as \( x \to 5^+ \) is:
  \[
  \lim_{x \to 5^+} f(x) = 13 - 2(5) = 13 - 10 = 3
  \]

### 3. Evaluating \( f(5) \)
- Since \( x = 5 \) falls into the second piece of the function (where \( x \geq 5 \)), we use \( f(x) = 13 - 2x \):
  \[
  f(5) = 13 - 2(5) = 13 - 10 = 3
  \]

### 4. Determining Continuity at \( x = 5 \)
For \( f(x) \) to be continuous at \( x = 5 \), the following must hold:
- The left-hand limit \( \lim_{x \to 5^-} f(x) \),
- The right-hand limit \( \lim_{x \to 5^+} f(x) \),
- And the function value \( f(5) \) must all be equal.

In this case:
- \( \lim_{x \to 5^-} f(x) = 14 \),
- \( \lim_{x \to 5^+} f(x) = 3 \),
- \( f(5) = 3 \).

Since the left-hand limit and right-hand limit are not equal, the function is **not continuous** at \( x = 5 \).

### Final Answers:
- \( \lim_{x \to 5^-} f(x) = 14 \)
- \( \lim_{x \to 5^+} f(x) = 3 \)
- \( f(5) = 3 \)
- Is \( f \) continuous at \( x = 5 \)? **NO**

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:
1. What conditions must be satisfied for a function to be continuous at a point?
2. Can a function be continuous at a point if its left-hand and right-hand limits are unequal?
3. How does a jump discontinuity differ from a removable discontinuity?
4. What are the key differences between piecewise functions and normal functions in terms of continuity?
5. Can the graph of this function help visually confirm its discontinuity at \( x = 5 \)?

**Tip:** Always check both one-sided limits when evaluating continuity at a specific point, especially in piecewise functions!

Find the indicated one-sided limits of f(x) and determine the continuity of f at the indicated point. The function is given as:
 f(x) = { 4 + 2x, x < 5; 13 - 2x, x >= 5 }

Solve a piecewise function's one-sided limits and check continuity at x = 5. Includes step-by-step solutions for finding left-hand and right-hand limits, as well as evaluating the function's value and determining if the function is continuous. Suitable for Grades 11-12.

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