The given limit is:

$\lim_{h \to 0} \frac{\sqrt{25 + h} - 5}{h}$

This expression is in the form of the definition of a derivative:

$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

From this, we can infer the function $f(x)$ and the value of $a$.

Step 1: Identify $f(x)$

By comparing the structure, the term $\sqrt{25 + h}$ suggests that $f(x) = \sqrt{x}$, because it's the expression inside the limit after substituting $x = 25$. This form appears when calculating the derivative of a square root function.

Step 2: Identify $a$

Next, we need to identify $a$. The term $\sqrt{25} = 5$, and in the limit expression, we have $5$ in the denominator subtraction. Therefore, $a = 25$, because $f(25) = 5$.

Conclusion:

• $f(x) = \sqrt{x}$
• $a = 25$

Would you like a detailed explanation of the derivative process or any other aspect of the problem?

Related Questions:

1. How do you derive the formula for the derivative of $\sqrt{x}$?
2. What is the derivative of $\sqrt{x}$ at any point $x$?
3. How can we evaluate limits using rationalization techniques?
4. What are other common forms of limits that define derivatives?
5. What is the geometric interpretation of the derivative in this context?

Tip: Rationalizing the numerator is often helpful when dealing with limits involving square roots.