1. **Problem (a):** Evaluate $$\lim_{(x,y) \to (1,1)} \frac{x^2 y - xy^2}{x - y}$$ Step 1: Factor the numerator: $$x^2 y - xy^2 = xy(x - y)$$ Step 2: Substitute the factorization into the limit expression: $$\frac{xy(x - y)}{x - y}$$ Step 3: Cancel the common factor $(x - y)$ (valid since we consider the limit, not the point where $x=y$): $$xy$$ Step 4: Evaluate the limit by substituting $(x,y) = (1,1)$: $$1 \times 1 = 1$$ **Answer (a):** The limit is $1$. 2. **Problem (b):** Evaluate $$\lim_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + 2xy + y^2}$$ Step 1: Factor numerator and denominator: - Numerator: $$x^2 - y^2 = (x - y)(x + y)$$ - Denominator: $$x^2 + 2xy + y^2 = (x + y)^2$$ Step 2: Substitute factorizations: $$\frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}$$ Step 3: Evaluate the limit as $(x,y) \to (0,0)$: - The expression simplifies to $$\frac{x - y}{x + y}$$ - Approaching along $y = x$, numerator and denominator both zero, but limit depends on path. Step 4: Check limit along different paths: - Along $y = x$: $$\frac{x - x}{x + x} = \frac{0}{2x} = 0$$ - Along $y = -x$: $$\frac{x - (-x)}{x + (-x)} = \frac{2x}{0}$$ which is undefined. **Answer (b):** The limit does not exist because it depends on the path. 3. **Problem (c):** Evaluate $$\lim_{(x,y) \to (4,4)} \frac{\sqrt{x} - \sqrt{y}}{x - y}$$ Step 1: Rationalize the numerator: $$\frac{\sqrt{x} - \sqrt{y}}{x - y} \times \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}} = \frac{x - y}{(x - y)(\sqrt{x} + \sqrt{y})} = \frac{1}{\sqrt{x} + \sqrt{y}}$$ Step 2: Evaluate the limit by substituting $(x,y) = (4,4)$: $$\frac{1}{\sqrt{4} + \sqrt{4}} = \frac{1}{2 + 2} = \frac{1}{4}$$ **Answer (c):** The limit is $\frac{1}{4}$. **Summary:** - (a) Limit is $1$. - (b) Limit does not exist. - (c) Limit is $\frac{1}{4}$.