1. The problem asks to find the limit $$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1}$$. 2. This limit is of the form $$\frac{f(x) - f(a)}{x - a}$$ where $$f(x) = \sqrt[3]{x}$$ and $$a = 1$$. 3. Such a limit represents the derivative of $$f(x)$$ at $$x = a$$, so $$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1} = f'(1)$$. 4. The derivative of $$f(x) = x^{1/3}$$ is $$f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}$$. 5. Evaluate the derivative at $$x = 1$$: $$f'(1) = \frac{1}{3 \sqrt[3]{1^2}} = \frac{1}{3}$$. 6. Therefore, the limit is $$\boxed{\frac{1}{3}}$$.