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} -