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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.

Learn how to find the limit of the expression (-3 + h)^2 - 9 / h as h approaches 0 with a detailed step-by-step solution. This problem involves limits, derivatives, and algebraic simplification. Suitable for advanced high school students.

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