Step 1 :First, we need to find an expression that is equal to \(f(x, y)=\frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}}\), for all \((x, y)\) in the domain of \(f\), that will be better suited to find the limit, if it exists.
Step 2 :We can use the difference of cubes formula, \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\), to rewrite the numerator and denominator. Here, \(a = x^{\frac{1}{3}}\) and \(b = y^{\frac{1}{3}}\).
Step 3 :So, \(a^3 - b^3 = x - y\) and \(a^2+ab+b^2 = x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}\).
Step 4 :Then, \(f(x, y) = \frac{x - y}{x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}}\).
Step 5 :So, the answer to the first part is A. \(\frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}} = \frac{x - y}{x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}}\) for all \((x, y)\) in the domain of \(f\).
Step 6 :Next, we need to evaluate the limit. \(\lim _{(x, y) \rightarrow(729,729)} \frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}}\).
Step 7 :Substitute \((x, y) = (729, 729)\) into the expression we just found.
Step 8 :We get \(\frac{729 - 729}{729^{\frac{2}{3}} + 729^{\frac{1}{3}}*729^{\frac{1}{3}} + 729^{\frac{2}{3}}} = \frac{0}{3*729^{\frac{2}{3}}} = 0\).
Step 9 :So, the answer to the second part is A. \(\lim _{(x, y) \rightarrow(729,729)} \frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}} = \boxed{0}\).