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The given limit problem is:

\[
\lim_{h \to 0} \frac{(h + 4)^2 - 16}{h}
\]

### Step-by-Step Solution:

1. **Expand** the term \((h + 4)^2\):
   \[
   (h + 4)^2 = h^2 + 8h + 16
   \]

2. Substitute this into the limit expression:
   \[
   \lim_{h \to 0} \frac{(h^2 + 8h + 16) - 16}{h}
   \]

3. Simplify the numerator:
   \[
   (h^2 + 8h + 16) - 16 = h^2 + 8h
   \]
   So the limit becomes:
   \[
   \lim_{h \to 0} \frac{h^2 + 8h}{h}
   \]

4. **Factor** out \(h\) from the numerator:
   \[
   \frac{h^2 + 8h}{h} = h + 8
   \]

5. Now evaluate the limit:
   \[
   \lim_{h \to 0} (h + 8) = 8
   \]

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

Let me know if you'd like further details or have any questions!

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Here are 5 related questions to expand this concept:

1. How would you evaluate the limit \(\lim_{h \to 0} \frac{(h + a)^2 - a^2}{h}\) for any constant \(a\)?
2. Can you apply the same limit technique to \(\lim_{h \to 0} \frac{(h + 3)^3 - 27}{h}\)?
3. How do you determine if a function is continuous at a point using limits?
4. What is the significance of limits in the definition of the derivative?
5. How does the binomial expansion help in solving limits involving powers?

**Tip:** When solving limits, always check if direct substitution gives an indeterminate form (like \(\frac{0}{0}\)) before applying algebraic techniques like factoring or expanding.

Evaluate the limit: lim(h → 0) [(h + 4)^2 - 16] / h

Learn how to solve the limit lim(h → 0) [(h + 4)^2 - 16] / h using algebraic manipulation and limit evaluation techniques. This step-by-step guide covers key concepts like factoring and limit theorems. Suitable for students in Grades 11-12.

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