The given problem involves finding a limit as $h$ approaches 0 for the following expression:

$\lim_{h \to 0} \frac{(-3 + h)^2 - 9}{h}$

This expression resembles the definition of a derivative. Let me solve it step by step.

Step 1: Expand the expression

First, expand $(-3 + h)^2$:

$(-3 + h)^2 = 9 - 6h + h^2$

Step 2: Substitute and simplify the expression

Now, substitute $(-3 + h)^2$ into the limit expression:

$\frac{9 - 6h + h^2 - 9}{h}$

Simplifying the numerator:

$= \frac{-6h + h^2}{h}$

Step 3: Factor and simplify further

Factor the numerator:

$= \frac{h(-6 + h)}{h}$

Cancel out $h$ (as $h \neq 0$ for simplification purposes):

$= -6 + h$

Step 4: Evaluate the limit

Finally, evaluate the limit as $h \to 0$:

$\lim_{h \to 0} (-6 + h) = -6$

Final Answer:

$\lim_{h \to 0} \frac{(-3 + h)^2 - 9}{h} = -6$

Would you like a detailed explanation of the derivative concept or have any other questions?

Here are five related questions:

1. How does this limit relate to the concept of a derivative?
2. What is the geometric interpretation of this limit?
3. Can you derive the general formula for the derivative using limits?
4. How would this problem change if the function involved $x^3$ instead of a square?
5. What happens if the value of the constant 9 is changed to another number?

Tip: Whenever you're working with limits involving small changes, it’s useful to recognize patterns related to derivatives.