https://file.aiquickdraw.com/mathbot/file/17310170933030iwkg8am.jpg

To solve the limit problem, let’s analyze it step-by-step. The expression is:

\[
\lim_{{x \to 12}} \frac{x^2 + 3x - 180}{x^2 - 144}
\]

### Step 1: Substitute \( x = 12 \)

By direct substitution, we get:

\[
\frac{12^2 + 3 \cdot 12 - 180}{12^2 - 144} = \frac{144 + 36 - 180}{144 - 144} = \frac{0}{0}
\]

Since this yields an indeterminate form (\( \frac{0}{0} \)), we need to simplify the expression further.

### Step 2: Factor the Numerator and Denominator

1. **Numerator:** \( x^2 + 3x - 180 \)
   - To factor this, we look for two numbers that multiply to \(-180\) and add up to \(3\).
   - The factors are \(15\) and \(-12\), so we can write:
     \[
     x^2 + 3x - 180 = (x + 15)(x - 12)
     \]

2. **Denominator:** \( x^2 - 144 \)
   - This is a difference of squares:
     \[
     x^2 - 144 = (x + 12)(x - 12)
     \]

### Step 3: Simplify the Expression

The limit now becomes:

\[
\lim_{{x \to 12}} \frac{(x + 15)(x - 12)}{(x + 12)(x - 12)}
\]

We can cancel out the \( (x - 12) \) terms from the numerator and the denominator, assuming \( x \neq 12 \):

\[
\lim_{{x \to 12}} \frac{x + 15}{x + 12}
\]

### Step 4: Substitute \( x = 12 \) Again

Now that the expression is simplified, we substitute \( x = 12 \):

\[
\frac{12 + 15}{12 + 12} = \frac{27}{24} = \frac{9}{8}
\]

### Conclusion

The limit is:

\[
\lim_{{x \to 12}} \frac{x^2 + 3x - 180}{x^2 - 144} = \frac{9}{8}
\]

Would you like further details on any part of the solution or have any other questions?

Here are five related questions for further exploration:

1. How can we apply L'Hôpital's Rule to solve indeterminate forms?
2. What are other techniques for simplifying limits involving polynomials?
3. How can limits help us understand continuity at a given point?
4. How does the concept of the difference of squares help in simplifying expressions?
5. What are some common factorization techniques useful in limit problems?

**Tip:** When you encounter a \(\frac{0}{0}\) indeterminate form, consider factoring or simplifying the expression before trying other methods like L'Hôpital's Rule.

Determine the limit: lim (x→12) (x^2 + 3x - 180) / (x^2 - 144).

Explore how to determine the limit of the rational function lim (x→12) (x^2 + 3x - 180) / (x^2 - 144). Learn key algebraic techniques such as factorization and simplification to find the limit. Suitable for high school math students (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