how do i approach this: if f(x) = sqrt(x+2) then lim h->0 (f(2+h)-f(2))/h =

This is a limit that involves the difference quotient, and it is essentially the definition of the derivative of the function \( f(x) \) at the point \( x = 2 \). To approach this limit, we can follow these steps:

1. **Recall the definition of the derivative**:  
   The limit given is the definition of the derivative of \( f(x) \) at \( x = 2 \):
   \[
   \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}
   \]
   This is \( f'(2) \), the derivative of \( f(x) \) at \( x = 2 \).

2. **Find the derivative of \( f(x) \)**:  
   The function \( f(x) = \sqrt{x+2} \). To find the derivative \( f'(x) \), we use the chain rule:
   \[
   f'(x) = \frac{d}{dx} \left( \sqrt{x+2} \right) = \frac{1}{2\sqrt{x+2}} \cdot 1
   \]
   Therefore, \( f'(x) = \frac{1}{2\sqrt{x+2}} \).

3. **Evaluate the derivative at \( x = 2 \)**:  
   Now, substitute \( x = 2 \) into the derivative:
   \[
   f'(2) = \frac{1}{2\sqrt{2+2}} = \frac{1}{2\sqrt{4}} = \frac{1}{2 \times 2} = \frac{1}{4}
   \]

Thus, the value of the limit is:
\[
\lim_{h \to 0} \frac{f(2+h) - f(2)}{h} = f'(2) = \frac{1}{4}
\]

### Do you want further details or have any questions?

Here are 5 related questions that expand on this concept:
1. How would you find the derivative of \( f(x) = x^n \) using the limit definition?
2. What is the general form of the derivative of \( f(x) = \sqrt{x + c} \)?
3. How does the chain rule apply to composite functions like \( f(x) = \sqrt{g(x)} \)?
4. How do you approach the limit definition of the derivative when the function involves a fraction?
5. What is the geometric interpretation of the derivative at a point?

**Tip**: When working with square root functions, remember to be careful with the chain rule since the derivative depends on the inner function as well!

Understand how to approach finding the derivative using the limit definition with a specific example. Learn step-by-step how to calculate the derivative of f(x) = sqrt(x+2) at x = 2. Suitable for advanced high school and college-level students.

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