To solve the given limit problem:

$\lim_{x \to 3} \frac{x^3 - x^2 - 18}{x - 3}$

Steps:

1. Direct Substitution: Substituting $x = 3$ into the numerator and denominator: $\text{Numerator: } 3^3 - 3^2 - 18 = 27 - 9 - 18 = 0$ $\text{Denominator: } 3 - 3 = 0$ This results in an indeterminate form $\frac{0}{0}$, so we apply factoring or simplification.

2. Factorize the Numerator: The numerator $x^3 - x^2 - 18$ can be factored. First, try synthetic or polynomial division by $(x - 3)$.

   Perform synthetic division of $x^3 - x^2 - 18$ by $(x - 3)$:

   • Divide $x^3 - x^2 - 18$ by $(x - 3)$, the quotient is: $x^2 + 2x + 6$

   So, $x^3 - x^2 - 18 = (x - 3)(x^2 + 2x + 6)$.

3. Simplify the Expression: Substitute the factorization into the original fraction: $\frac{x^3 - x^2 - 18}{x - 3} = \frac{(x - 3)(x^2 + 2x + 6)}{x - 3}$ Cancel the $(x - 3)$ term: $= x^2 + 2x + 6$

4. Evaluate the Limit: Substitute $x = 3$ into the simplified expression: $\lim_{x \to 3} (x^2 + 2x + 6) = 3^2 + 2(3) + 6 = 9 + 6 + 6 = 21$

Final Answer:

$\boxed{21}$

Let me know if you'd like more clarification or further examples! Here are some related questions:

1. What is synthetic division, and how does it help simplify rational expressions?
2. Can the numerator be factored further to solve limits of a similar type?
3. Why does the $\frac{0}{0}$ form indicate the need for further simplification in limits?
4. How is the Factor Theorem used to check divisibility of polynomials?
5. What other methods can be used to solve limits involving indeterminate forms?

Tip: Always check for factoring opportunities in indeterminate forms like $\frac{0}{0}$ to simplify the limit.