\[ \begin{align} \lim_{x\rightarrow a}\;{\frac{x^{2}-(a+1)x+a}{x^{3}-a^{3}}} &=\lim_{x\rightarrow a}\;{\frac{x^{2}-ax-x+a}{(x-a)(x^{2}+ax+a^{2})}}=\\[5pt] &=\lim_{x\rightarrow a}\;{\frac{(x^{2}-x)-(ax-a)}{(x-a)(x^{2}+ax+a^{2})}}=\\[5pt] &=\lim_{x\rightarrow a}\;{\frac{x(x-1)-a(x-1)}{(x-a)(x^{2}+ax+a^{2})}}=\\[5pt] &=\lim_{x\rightarrow a}\;{\frac{\cancel{(x-a)}(x-1)}{\cancel{(x-a)}(x^{2}+ax+a^{2})}}=\\[5pt] &=\lim_{x\rightarrow a}\;{\frac{(x-1)}{(x^{2}+ax+a^{2})}}=\\[5pt] &=\lim_{x\rightarrow a}\;{\frac{(a-1)}{(a^{2}+a.a+a^{2})}}=\frac{a-1}{3a^{2}} \end{align} \]

\[ \bbox[#FFCCCC,10px] {\lim_{x\rightarrow a}\;{\frac{x^{2}-(a+1)x+a}{x^{3}-a^{3}}}=\frac{a-1}{3a^{2}}} \]