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

To solve the limit \(\lim_{{x \to 8}} \frac{\sqrt[3]{x} - 2}{x - 8}\), let's analyze it step by step.

### Step 1: Direct Substitution
First, try to directly substitute \( x = 8 \) 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 apply a different method to evaluate the limit.

### Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that if the limit results in an indeterminate form \( \frac{0}{0} \), you can differentiate the numerator and the denominator separately and then take the limit.

1. Differentiate the numerator: \(\sqrt[3]{x} - 

Learn how to evaluate the limit \(\lim_{{x \to 8}} \frac{{\sqrt[3]{{x}} - 2}}{{x - 8}}\) using L'Hôpital's Rule. This problem involves understanding limits and applying advanced calculus techniques. Suitable for advanced high school students and calculus enthusiasts.

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