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The limit in the given problem is:

\[
\lim_{x \to 25} \frac{5\sqrt{x} - 25}{x - 25}
\]

This limit represents \( f'(c) \) where \( c = 25 \). Our goal is to find the function \( f(x) \) and then use the value \( c = 25 \) to compute \( f'(25) \).

### Step-by-Step Solution:

1. **Interpret the limit as a derivative:**
   The expression \(\lim_{x \to 25} \frac{f(x) - f(25)}{x - 25}\) is the definition of the derivative at \( x = 25 \). Hence, from the given limit, we can identify \( f(x) = 5\sqrt{x} \).

2. **Verification of \( f(x) \):**
   We know that:
   \[
   f(x) = 5\sqrt{x}
   \]
   Now, \( f(25) = 5\sqrt{25} = 25 \), so the limit simplifies to:
   \[
   \lim_{x \to 25} \frac{5\sqrt{x} - 25}{x - 25}
   \]
   which matches the given limit exactly.

Thus, the function \( f(x) \) is:

\[
f(x) = 5\sqrt{x}
\]

### Answer:
- \( f(x) = 5\sqrt{x} \)
- \( c = 25 \) (already provided)

Would you like to see more details or explanations on the steps?

Here are 5 related questions to explore:

1. How do you compute the derivative of \( f(x) = 5\sqrt{x} \)?
2. What are the alternative ways to solve limits involving radicals?
3. Can you explain how to use the chain rule to differentiate \( f(x) = 5\sqrt{x} \)?
4. How would the derivative change if the limit involved \( \sqrt{x+1} \) instead?
5. How does L'Hopital's Rule apply to this limit problem?

**Tip:** When dealing with limits that result in an indeterminate form \( \frac{0}{0} \), consider using factoring, rationalizing, or L'Hopital's Rule.

The limit represents f'(c) for a function f(x) and a number c. Find f(x) and c. The limit is: lim x->25 (5√x - 25) / (x - 25).

This problem asks to find the function f(x) and the value of c given the limit lim x->25 (5√x - 25)/(x - 25), which represents the derivative of f(x) at c = 25. Learn how to interpret limits as derivatives and identify the function based on the provided limit.

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