https://file.aiquickdraw.com/mathbot/file/1729889027303h6prf9f3.jpg

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.

The limit \( \lim_{h \to 0} \frac{\sqrt{25 + h} - 5}{h} \) is the standard expression defining the derivative of some function f(x) at some number a. Find f and a.

Learn how to solve a calculus problem involving the limit \( \lim_{h \to 0} \frac{\sqrt{25 + h} - 5}{h} \), and find the function f(x) and value a. This problem demonstrates the application of the limit definition of a derivative.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation