1. **State the problem:** Find the limit as $x$ approaches 1 of the expression $$\frac{1 - x^{\frac{2}{3}}}{1 - x^{\frac{1}{3}}}.$$\n\n2. **Recall the formula and rules:** When evaluating limits that result in an indeterminate form like $\frac{0}{0}$, we can try to simplify the expression algebraically. Here, notice that $x^{\frac{2}{3}} = \left(x^{\frac{1}{3}}\right)^2$. Let $t = x^{\frac{1}{3}}$. Then the limit becomes $$\lim_{t \to 1} \frac{1 - t^2}{1 - t}.$$\n\n3. **Simplify the expression:** The numerator $1 - t^2$ factors as $(1 - t)(1 + t)$. So, $$\frac{1 - t^2}{1 - t} = \frac{(1 - t)(1 + t)}{1 - t}.$$\n\n4. **Cancel common factors:** We can cancel $1 - t$ in numerator and denominator, but since $t \to 1$, $1 - t \neq 0$ in the limit process (except at the point), so cancellation is valid: $$\frac{\cancel{(1 - t)}(1 + t)}{\cancel{1 - t}} = 1 + t.$$\n\n5. **Evaluate the limit:** Substitute $t = 1$ back: $$1 + 1 = 2.$$\n\n**Final answer:** $$\boxed{2}.$$