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The given expression is:

\[
\lim_{\Delta x \to 0} \frac{(x + \Delta x)^3 - x^3}{\Delta x}
\]

To solve this limit using the identity \((a - b)(a^2 + ab + b^2)\), we first recognize that the expression \((x + \Delta x)^3 - x^3\) is a difference of cubes. 

We can apply the difference of cubes factorization:

\[
(a^3 - b^3) = (a - b)(a^2 + ab + b^2)
\]

Here, \(a = x + \Delta x\) and \(b = x\). Substituting these into the difference of cubes formula, we get:

\[
(x + \Delta x)^3 - x^3 = \left[(x + \Delta x) - x\right]\left[(x + \Delta x)^2 + (x + \Delta x)x + x^2\right]
\]

Simplifying the first factor:

\[
(x + \Delta x) - x = \Delta x
\]

So, the expression becomes:

\[
(x + \Delta x)^3 - x^3 = \Delta x \left[(x + \Delta x)^2 + (x + \Delta x)x + x^2\right]
\]

Now, substitute this into the limit expression:

\[
\lim_{\Delta x \to 0} \frac{\Delta x \left[(x + \Delta x)^2 + (x + \Delta x)x + x^2\right]}{\Delta x}
\]

The \(\Delta x\) in the numerator and denominator cancels out:

\[
\lim_{\Delta x \to 0} \left[(x + \Delta x)^2 + (x + \Delta x)x + x^2\right]
\]

Now, substitute \(\Delta x = 0\) in the remaining expression:

\[
\lim_{\Delta x \to 0} \left[x^2 + x^2 + x^2\right] = 3x^2
\]

So, the final answer is:

\[
\lim_{\Delta x \to 0} \frac{(x + \Delta x)^3 - x^3}{\Delta x} = 3x^2
\]

This result represents the derivative of \(x^3\) with respect to \(x\), which confirms the calculation.

Do you need a further breakdown or have any questions?

Here are 5 related questions:
1. How would you compute the derivative of \(x^4\) using the limit definition?
2. What is the difference between the sum of cubes and difference of cubes?
3. How can you apply this method to find the derivative of \(x^n\)?
4. How does this factorization simplify the process of taking derivatives?
5. Can the difference of cubes identity be used for functions other than polynomials?

**Tip:** When dealing with limits involving polynomials, recognizing patterns like the difference of cubes can greatly simplify the process.

Learn how to solve limits using the difference of cubes identity with a detailed step-by-step solution. This problem covers key concepts in calculus, including limits, derivatives, and algebraic identities. Suitable for advanced high school students.

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