1. **State the problem:** Find the limit $$\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x}$$ using three different methods. 2. **Method 1: Recognize the limit as a derivative** This limit resembles the definition of the derivative of the function $$f(t) = t^{2/3}$$ at the point $$t = x$$: $$\lim_{y \to x} \frac{f(y) - f(x)}{y - x} = f'(x)$$ We can find $$f'(t)$$ using differentiation rules: $$f(t) = t^{2/3} = e^{\frac{2}{3}\ln t}$$ Differentiating: $$f'(t) = \frac{2}{3} t^{-1/3}$$ Therefore, $$\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x} = f'(x) = \frac{2}{3} x^{-1/3}$$ 3. **Method 2: Use algebraic manipulation (difference of powers)** Rewrite numerator using a factoring pattern related to fractional powers. Set $$a = y^{1/3}$$ and $$b = x^{1/3}$$, so: $$y^{2/3} - x^{2/3} = a^2 - b^2 = (a - b)(a + b)$$ Substitute back: $$\frac{y^{2/3} - x^{2/3}}{y - x} = \frac{(y^{1/3} - x^{1/3})(y^{1/3} + x^{1/3})}{y - x}$$ We know: $$y - x = (y^{1/3} - x^{1/3})(y^{2/3} + y^{1/3}x^{1/3} + x^{2/3})$$ So: $$\frac{y^{2/3} - x^{2/3}}{y - x} = \frac{(y^{1/3} - x^{1/3})(y^{1/3} + x^{1/3})}{(y^{1/3} - x^{1/3})(y^{2/3} + y^{1/3}x^{1/3} + x^{2/3})} = \frac{y^{1/3} + x^{1/3}}{y^{2/3} + y^{1/3}x^{1/3} + x^{2/3}}$$ Taking the limit $$y \to x$$: $$= \frac{x^{1/3} + x^{1/3}}{x^{2/3} + x^{2/3} + x^{2/3}} = \frac{2 x^{1/3}}{3 x^{2/3}} = \frac{2}{3} x^{-1/3}$$ 4. **Method 3: Use the definition of derivative via the binomial expansion for fractional powers** Expand $$y^{2/3}$$ near $$x$$ using the binomial approximation: Let $$y = x + h$$, with $$h \to 0$$. $$y^{2/3} = (x + h)^{2/3} \approx x^{2/3} + \frac{2}{3} x^{-1/3} h$$ Thus: $$\frac{y^{2/3} - x^{2/3}}{y - x} = \frac{x^{2/3} + \frac{2}{3} x^{-1/3} h - x^{2/3}}{h} = \frac{\frac{2}{3} x^{-1/3} h}{h} = \frac{2}{3} x^{-1/3}$$ **Final Answer:** $$\boxed{\lim_{y \to x} \frac{y^{2/3} - x^{2/3}}{y - x} = \frac{2}{3} x^{-1/3}}$$