To find the function $f(x)$ given its derivative in the form of a limit, we start with the expression:

$f'(x) = \lim_{h \to 0} \frac{\sqrt[3]{8 + 6h + h^2} - 2}{h}$

This expression resembles the definition of the derivative using limits, suggesting it might be the derivative of a function evaluated at a specific point. Let's evaluate this limit step-by-step to identify the function $f(x)$.

Step 1: Recognize the Expression

The expression inside the limit resembles the form of the derivative of the cube root function, $\sqrt[3]{x}$. More specifically, it looks like the derivative of $f(x) = \sqrt[3]{x}$ at some point.

Step 2: Identify the Point

The expression $\sqrt[3]{8 + 6h + h^2}$ suggests that it is centered around the point where $x = 8$. This is because when $h = 0$, the cube root expression becomes $\sqrt[3]{8} = 2$.

Step 3: Set up the General Derivative Formula

The derivative of the cube root function, $f(x) = \sqrt[3]{x}$, is:

$f'(x) = \frac{1}{3} x^{-\frac{2}{3}}$

Let's evaluate this derivative at $x = 8$:

$f'(8) = \frac{1}{3} \times 8^{-\frac{2}{3}} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$

Step 4: Evaluate the Limit

We have previously identified that the given limit evaluates the derivative at the point $x = 8$, and the limit should match this evaluation:

$\lim_{h \to 0} \frac{\sqrt[3]{8 + 6h + h^2} - 2}{h} = \frac{1}{12}$

Thus, the function $f(x)$ whose derivative gives the limit expression is:

$f(x) = \sqrt[3]{x}$

The derivative calculation above confirms that this matches the given limit, as the derivative of $\sqrt[3]{x}$ evaluated at $x = 8$ gives $\frac{1}{12}$.

Conclusion

Therefore, the function $f(x)$ is:

$f(x) = \sqrt[3]{x}$

Would you like to see more details, or do you have any questions about this explanation?

---

Here are some additional related questions you might find interesting:

1. What is the derivative of $f(x) = \sqrt[3]{x}$ at $x = 1$?
2. How do you differentiate $f(x) = x^{1/3}$ using the power rule?
3. Can you find the derivative of $f(x) = \sqrt[3]{x^3 + 1}$?
4. What is the derivative of $g(x) = \sqrt[3]{x^2}$?
5. How can you approximate $\sqrt[3]{9}$ using the derivative of $f(x) = \sqrt[3]{x}$?
6. What are the critical points of $f(x) = \sqrt[3]{x}$?
7. How does the graph of $f(x) = \sqrt[3]{x}$ compare to $g(x) = x^{1/3}$?
8. How does the derivative of $f(x) = \sqrt[3]{x}$ help in understanding its concavity and inflection points?

Tip: To differentiate functions involving roots, consider rewriting them as powers, such as $x^{1/3}$, to apply the power rule easily.