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!