We are tasked with finding the limit:

\[
\lim_{h \to 0} \frac{(8+h)^{1/3} - 2}{h}
\]

This is a typical problem involving limits and derivatives. Specifically, this limit can be interpreted as the derivative of the function \( f(x) = x^{1/3} \) at \( x = 8 \).

### Solution:

We need to evaluate this limit using the following steps:

1. **Recognize the form**: The expression resembles the definition of a derivative. It can be written as:

\[
f'(8) = \lim_{h \to 0} \frac{f(8+h) - f(8)}{h}
\]

where \( f(x) = x^{1/3} \) and \( f(8) = 2 \) since \( 8^{1/3} = 2 \).

2. **Differentiate \( f(x) = x^{1/3} \)**:
   Using the power rule for derivatives \( \frac{d}{dx}(x^n) = n x^{n-1} \), the derivative of \( f(x) = x^{1/3} \) is:

\[
f'(x) = \frac{1}{3} x^{-2/3}
\]

3. **Evaluate the derivative at \( x = 8 \)**:

\[
f'(8) = \frac{1}{3} \cdot 8^{-2/3}
\]

We know that \( 8^{1/3} = 2 \), so:

\[
f'(8) = \frac{1}{3} \cdot \frac{1}{2^2} = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}
\]

### Conclusion:

Thus, the value of the limit is:

\[
\lim_{h \to 0} \frac{(8+h)^{1/3} - 2}{h} = \frac{1}{12}
\]

---

Would you like more details on any of the steps, or do you have any further questions?

Here are 5 related questions to explore further:
1. What is the derivative of \( f(x) = x^{n} \) for other powers of \( n \)?
2. How can limits be used to calculate the slope of a curve at a point?
3. What are some other real-world applications of the derivative?
4. How does the limit definition of a derivative apply to other functions?
5. What is the geometric interpretation of a derivative at a given point?

**Tip**: When evaluating limits, especially those resembling the definition of the derivative, recognizing the pattern can save significant calculation time.

Explore the solution to the limit problem lim h->0 ((8+h)^1/3 - 2)/h, which involves finding the derivative of x^(1/3) at x = 8. Learn key calculus concepts, including limits, derivatives, and the power rule, suitable for Grades 11-12.

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