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To solve this problem, we need to determine the value of \( k \) such that the function \( g(x) \) is continuous for all \( x > -13 \).

### Step 1: Understand the given function
The function \( g(x) \) is defined as:

\[
g(x) = \begin{cases} 
\frac{x - 3}{\sqrt{x + 13} - 4} & \text{for } x \geq -13, x \neq 3 \\
k & \text{for } x = 3
\end{cases}
\]

The function is continuous for all \( x > -13 \). For \( g(x) \) to be continuous at \( x = 3 \), the value of \( g(x) \) as \( x \to 3 \) (from the piecewise function definition) must equal \( g(3) = k \).

### Step 2: Find the limit as \( x \to 3 \)
To make \( g(x) \) continuous at \( x = 3 \), we must find the limit of \( \frac{x - 3}{\sqrt{x + 13} - 4} \) as \( x \to 3 \).

Substituting \( x = 3 \) into the function directly gives a \( \frac{0}{0} \) indeterminate form, so we apply algebraic simplification.

#### Simplifying the expression:
\[
\lim_{x \to 3} \frac{x - 3}{\sqrt{x + 13} - 4}
\]

Multiply the numerator and denominator by the conjugate of the denominator:
\[
\lim_{x \to 3} \frac{(x - 3)(\sqrt{x + 13} + 4)}{(\sqrt{x + 13} - 4)(\sqrt{x + 13} + 4)}
\]

This simplifies to:
\[
\lim_{x \to 3} \frac{(x - 3)(\sqrt{x + 13} + 4)}{(x + 13) - 16}
= \lim_{x \to 3} \frac{(x - 3)(\sqrt{x + 13} + 4)}{x - 3}
\]

Cancel out the \( (x - 3) \) terms:
\[
\lim_{x \to 3} \sqrt{x + 13} + 4
\]

Now substitute \( x = 3 \) into the simplified expression:
\[
\sqrt{3 + 13} + 4 = \sqrt{16} + 4 = 4 + 4 = 8
\]

### Step 3: Set the limit equal to \( k \)
For \( g(x) \) to be continuous at \( x = 3 \), the value of \( k \) must equal the limit as \( x \to 3 \). Therefore, \( k = 8 \).

### Final Answer:
The value of \( k \) is \( \boxed{8} \).

---

Would you like further details or explanations on any step?

Here are some related questions to explore:
1. What does it mean for a function to be continuous at a point?
2. How do you use limits to test for continuity?
3. Why did we multiply by the conjugate in this problem?
4. Can you always simplify indeterminate forms like \( \frac{0}{0} \) using algebraic techniques?
5. How does the value of \( k \) affect the overall behavior of the function?

**Tip**: When you encounter a \( \frac{0}{0} \) indeterminate form, applying algebraic tricks like multiplying by the conjugate often helps to simplify the limit.

Let g(x) be a piecewise function defined as follows: g(x) = (x-3)/(sqrt(x+13)-4) for x ≥ -13, x ≠ 3, and g(x) = k for x = 3. The function g is continuous for all x > -13. What is the value of k?

Learn how to solve a piecewise function continuity problem by finding the value of k that ensures continuity at a specific point. This solution involves applying limits and algebraic simplification, making it ideal for high school students.

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