The given limit is:

$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$

Step 1: Factor the numerator

The numerator $x^2 - 9$ can be factored as a difference of squares:

$x^2 - 9 = (x - 3)(x + 3)$

Step 2: Simplify the expression

Substitute the factored form into the expression:

$\frac{(x - 3)(x + 3)}{x - 3}$

For $x \neq 3$, the terms $x - 3$ cancel out, leaving:

$x + 3$

Step 3: Evaluate the limit

Now, substitute $x = 3$ into the simplified expression:

$x + 3 = 3 + 3 = 6$

Thus, the limit is:

$\boxed{6}$

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the definition of a limit in calculus?
2. How do we deal with indeterminate forms like $\frac{0}{0}$?
3. How can limits be used to define the derivative of a function?
4. What are some common techniques for evaluating limits?
5. What is the difference between one-sided and two-sided limits?

Tip: Always check if factoring or other algebraic manipulations can simplify a limit expression before applying direct substitution.