Limit of ³√x minus 2 divided by x minus 8 as x approaches 8

To find the limit of \(\frac{\sqrt[3]{x} - 2}{x - 8}\) as \(x\) approaches 8, we'll follow these steps:

### Step 1: Direct Substitution
First, substitute \(x = 8\) directly into the expression:

\[
\frac{\sqrt[3]{8} - 2}{8 - 8} = \frac{2 - 2}{0} = \frac{0}{0}
\]

This results in an indeterminate form \(\frac{0}{0}\), so we need to use another method to evaluate the limit.

### Step 2: Rationalize the Numerator
To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator's cube root, using the identity:

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

Let \(a = \sqrt[3]{x}\) and \(b = 2\). Then:

\[
a^3 - b^3 = x - 8 = (\sqrt[3]{x} - 2)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)
\]

So,

\[
\frac{\sqrt[3]{x} - 2}{x - 8} = \frac{(\sqrt[3]{x} - 2)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)}{(x - 8)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)}
\]

Since \(x - 8 = (\sqrt[3]{x} - 2)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)\), the expression simplifies to:

\[
\frac{1}{\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4}
\]

### Step 3: Take the Limit
Now, substitute \(x = 8\) into the simplified expression:

\[
\lim_{x \to 8} \frac{1}{\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4} = \frac{1}{\sqrt[3]{8^2} + 2\sqrt[3]{8} + 4} = \frac{1}{\sqrt[3]{64} + 2 \times 2 + 4} = \frac{1}{4 + 4 + 4} = \frac{1}{12}
\]

### Final Answer
The limit is:

\[
\lim_{x \to 8} \frac{\sqrt[3]{x} - 2}{x - 8} = \frac{1}{12}
\]

Would you like any details or have further questions?

Here are five related questions you might find interesting:
1. How would you evaluate \(\lim_{x \to 1} \frac{x^3 - 1}{x - 1}\)?
2. Can L'Hôpital's Rule be used to solve this limit problem?
3. What is the geometric interpretation of limits in calculus?
4. How do you find the limit of a function as \(x\) approaches infinity?
5. What are other methods to resolve indeterminate forms besides rationalization?

**Tip:** When you encounter indeterminate forms, always consider algebraic manipulation, such as rationalization or factoring, before resorting to more advanced techniques.

Learn how to evaluate the limit of ∛x - 2 / (x - 8) as x approaches 8 with a step-by-step solution. This problem involves limits, rationalization, and resolving indeterminate forms in calculus. Suitable for undergraduate students and those preparing for calculus exams.

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