1. **Problem statement:** Find the limit $$\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x}.$$ 2. **Recognize the expression:** This is the difference quotient for the function $$f(t) = t^{2/3}$$ at $$t = x$$, which represents the derivative $$f'(x)$$ if the derivative exists. 3. **Rewrite the difference quotient:** $$\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x} = f'(x).$$ 4. **Differentiate the function:** Using the power rule for derivatives, $$f(t) = t^{2/3} \implies f'(t) = \frac{2}{3} t^{-1/3} = \frac{2}{3 t^{1/3}}.$$ 5. **Evaluate the limit:** Therefore, $$\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x} = f'(x) = \frac{2}{3 x^{1/3}}.$$ This is valid for $$x \neq 0$$ since the derivative involves division by $$x^{1/3}$$. **Final answer:** $$\boxed{\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x} = \frac{2}{3 x^{1/3}}}.$$