The given limit is:

$\lim_{{x \to 3}} \frac{x^2 - x - 12}{x^2 + 8x + 15}$

Step 1: Factor the numerator and the denominator

• Numerator: $x^2 - x - 12$ Factor it: $x^2 - x - 12 = (x - 4)(x + 3)$
• Denominator: $x^2 + 8x + 15$ Factor it: $x^2 + 8x + 15 = (x + 3)(x + 5)$

Step 2: Simplify the expression

Now the limit expression becomes: $\lim_{{x \to 3}} \frac{(x - 4)(x + 3)}{(x + 3)(x + 5)}$ Cancel the common factor $(x + 3)$: $\lim_{{x \to 3}} \frac{x - 4}{x + 5}$

Step 3: Evaluate the limit

Now substitute $x = 3$ into the simplified expression: $\frac{3 - 4}{3 + 5} = \frac{-1}{8}$

Thus, the limit is $\frac{-1}{8}$.

Correct Answer:

The correct answer is $\boxed{-\frac{1}{8}}$.

Would you like further details on any part of this solution? Here are 5 related questions you might explore:

1. How do you know when to factor in limits?
2. What happens if the numerator and denominator don't share common factors?
3. How do you handle limits where direct substitution leads to an undefined form?
4. How do you simplify rational expressions before applying the limit?
5. Can limits exist at points where there are vertical asymptotes?

Tip: Always check for factorable expressions when direct substitution gives an indeterminate form like $\frac{0}{0}$.