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:

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

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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.