Evaluate the limit.
\[
\lim _{(x, y) \rightarrow(729,729)} \frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}}
\]
Find an expression that is equal to $f(x, y)=\frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{\frac{2}{3}-y^{\frac{2}{3}}}$, for all $(x, y)$ in the domain of $f$, that will be better suited to find the limit, if it exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}}=\square$ for all $(x, y)$ in the domain of $f$. (Simplify your answer.)
B. The function $f(x, y)$ is already in a form suitable for finding the limit, if it exists.
Evaluate the limit. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\lim _{(x, y) \rightarrow(729,729)} \frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}}=\square$ (Simplify your answer.)
B. The limit does not exist.

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. \(\lim _{(x, y) \rightarrow(729,729)} \frac{x^{\frac{1}{3}}-y^{\frac{1}{3}}}{x^{\frac{2}{3}}-y^{\frac{2}{3}}} = \boxed{0}\).