The initial substitution of $x=a$ yields the form $\frac{0}{0}$. Simplify the function algebraically, or use a table or graph to determine the limit. If necessary, state that the limit does not exist.
\[
\lim _{x \rightarrow 3} \frac{x^{3}-27}{x-3}
\]

Step 1 ：The initial substitution of \(x=a\) yields the form \(\frac{0}{0}\). Simplify the function algebraically, or use a table or graph to determine the limit. If necessary, state that the limit does not exist.

Step 2 ：We are given the limit \(\lim _{x \rightarrow 3} \frac{x^{3}-27}{x-3}\).

Step 3 ：The function is undefined at \(x=3\), which is why we get the form \(\frac{0}{0}\). However, we can simplify the function by factoring the numerator.

Step 4 ：The expression \(x^{3}-27\) is a difference of cubes, which can be factored as \((x-3)(x^{2}+3x+9)\).

Step 5 ：After factoring, we can cancel out the \((x-3)\) terms in the numerator and denominator, which will allow us to evaluate the limit.

Step 6 ：The limit of the function as \(x\) approaches \(3\) is \(\boxed{27}\).