https://file.aiquickdraw.com/mathbot/file/1726943774585y8b9bfkq.jpg

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 \(\frac{0}{0}\).

Evaluate the limit: lim (x -> 3) (x^2 - x - 12) / (x^2 + 8x + 15). Select the correct answer from the given choices.

Learn how to evaluate the limit of a rational expression by factoring and simplifying. This step-by-step guide demonstrates how to solve the limit lim (x -> 3) (x^2 - x - 12) / (x^2 + 8x + 15) by canceling common factors, with detailed explanations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation