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To solve the limit 

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

we can follow these steps:

1. **Expand the numerator** \((2 + h)^4\):
   
   Using the binomial theorem, we have:
   \[
   (2 + h)^4 = 2^4 + 4 \cdot 2^3 \cdot h + 6 \cdot 2^2 \cdot h^2 + 4 \cdot 2 \cdot h^3 + h^4
   \]
   Simplifying, we get:
   \[
   (2 + h)^4 = 16 + 32h + 24h^2 + 8h^3 + h^4
   \]

2. **Subtract 16 from the expanded form**:
   \[
   (2 + h)^4 - 16 = (16 + 32h + 24h^2 + 8h^3 + h^4) - 16
   \]
   \[
   = 32h + 24h^2 + 8h^3 + h^4
   \]

3. **Rewrite the limit expression**:
   \[
   \lim_{{h \to 0}} \frac{{32h + 24h^2 + 8h^3 + h^4}}{h}
   \]

4. **Simplify the fraction by dividing each term by \(h\)**:
   \[
   = \lim_{{h \to 0}} (32 + 24h + 8h^3 + h^3)
   \]

5. **Evaluate the limit** as \(h\) approaches 0:
   \[
   \lim_{{h \to 0}} (32 + 24h + 8h^3 + h^3) = 32 + 24 \cdot 0 + 8 \cdot 0 + 0
   \]
   \[
   = 32
   \]

Therefore, the limit is:

\[
\boxed{32}
\]

Do you want details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the binomial theorem and how is it applied in this context?
2. How do you find the limit of a function as \( h \) approaches a specific value?
3. Can you explain why we can divide each term in the numerator by \( h \) in the limit expression?
4. How would the solution change if the power in the numerator was different, say \((2 + h)^3\)?
5. What are some common techniques for solving limits that involve polynomial expressions?
6. How can L'Hôpital's rule be used to solve limits involving indeterminate forms?
7. What is the geometric interpretation of finding the limit of a function?
8. How would you approach finding the limit if the numerator was a more complex function, such as a trigonometric function?

**Tip:** When dealing with limits, especially those that result in indeterminate forms like \(\frac{0}{0}\), try simplifying the expression or using algebraic techniques to rewrite the limit in a more manageable form.

Learn how to solve the limit problem (2 + h)^4 - 16 / h step-by-step with detailed explanations. This problem involves applying the binomial theorem and simplifying polynomial expressions. Suitable for advanced high school students and college-level learners.

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