https://file.aiquickdraw.com/mathbot/file/17267036193598vrjzev8.jpg

The problem you shared asks for the value of the following limit:

\[
\lim_{h \to 0} \frac{(16+h)^{\frac{1}{2}} - 2}{h}
\]

This expression is a derivative calculation in disguise. Let's break it down:

### Step-by-Step Explanation:

1. **Recognize the form**: 
   This limit is the definition of the derivative of the function \( f(x) = \sqrt{x} \) at \( x = 16 \). The formula for a derivative at a point \( a \) is:
   \[
   f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
   \]
   Here, \( f(x) = \sqrt{x} \), so \( f(16) = \sqrt{16} = 4 \).

2. **Apply the definition of the derivative**:
   \[
   \lim_{h \to 0} \frac{\sqrt{16+h} - 4}{h}
   \]
   This matches the form we are given in the problem, except the \( \sqrt{16+h} - 4 \) part is written as \( (16 + h)^{\frac{1}{2}} - 2 \) in the problem (since \( \sqrt{x} = x^{1/2} \) and \( \sqrt{16} = 4 \)).

3. **Evaluate the limit**:
   To solve the limit, we can rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator:
   \[
   \lim_{h \to 0} \frac{(16+h)^{\frac{1}{2}} - 4}{h} \cdot \frac{(16+h)^{\frac{1}{2}} + 4}{(16+h)^{\frac{1}{2}} + 4}
   \]
   Simplifying the numerator using the difference of squares:
   \[
   \lim_{h \to 0} \frac{(16+h) - 16}{h((16+h)^{\frac{1}{2}} + 4)} = \lim_{h \to 0} \frac{h}{h((16+h)^{\frac{1}{2}} + 4)}
   \]
   The \( h \)'s cancel out:
   \[
   \lim_{h \to 0} \frac{1}{(16+h)^{\frac{1}{2}} + 4}
   \]
   Now, substitute \( h = 0 \):
   \[
   \frac{1}{\sqrt{16} + 4} = \frac{1}{4 + 4} = \frac{1}{8}
   \]

### Final Answer:
\[
\lim_{h \to 0} \frac{(16+h)^{\frac{1}{2}} - 4}{h} = \frac{1}{8}
\]

Would you like further clarification or details? Here are some related questions to explore:

1. What is the general form of a derivative using limits?
2. How can rationalizing the numerator help in evaluating limits?
3. Why do we cancel out \( h \) in the fraction after rationalizing?
4. What is the significance of evaluating limits when \( h \to 0 \)?
5. How would this process change if the base value \( x = 16 \) were different?

**Tip:** Always check if a limit corresponds to the derivative definition. This helps simplify evaluation!

What is the value of lim(h → 0) [(16 + h)^(1/2) - 2] / h?

This problem asks for the limit of (16 + h)^(1/2) - 2 over h as h approaches 0, using concepts from calculus, including the derivative definition and algebraic manipulation. A step-by-step solution is provided, suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation