https://file.aiquickdraw.com/mathbot/file/1725668508738csrayo88.jpg

The limit problem shown in the image is:

\[
\lim_{{x \to 3}} \frac{x^3 + 9x^2 + 20x}{x^2 + x - 12}
\]

### Step 1: Factor the numerator and denominator (if possible).

**Numerator:** \(x^3 + 9x^2 + 20x\)
- Factor out \(x\):
  \[
  x(x^2 + 9x + 20)
  \]
- Now factor \(x^2 + 9x + 20\):
  \[
  x^2 + 9x + 20 = (x + 4)(x + 5)
  \]
- So, the numerator is:
  \[
  x(x + 4)(x + 5)
  \]

**Denominator:** \(x^2 + x - 12\)
- Factor the quadratic:
  \[
  x^2 + x - 12 = (x - 3)(x + 4)
  \]

### Step 2: Rewrite the expression with factored form.

The original expression becomes:
\[
\lim_{{x \to 3}} \frac{x(x + 4)(x + 5)}{(x - 3)(x + 4)}
\]

### Step 3: Simplify the expression.

- Cancel the common factor \((x + 4)\):
  \[
  \lim_{{x \to 3}} \frac{x(x + 5)}{x - 3}
  \]

### Step 4: Apply L'Hôpital's Rule or direct substitution.

At \(x = 3\), substituting directly into the simplified expression results in an indeterminate form \(0/0\), so we can either apply L'Hôpital's Rule or factor more.

Using **L'Hôpital's Rule**:

Differentiate the numerator and denominator:
- Derivative of \(x(x + 5)\): \(2x + 5\)
- Derivative of \(x - 3\): 1

Now substitute \(x = 3\) into the differentiated expression:
\[
\lim_{{x \to 3}} \frac{2x + 5}{1} = 2(3) + 5 = 6 + 5 = 11
\]

### Final Answer:
\[
\lim_{{x \to 3}} \frac{x^3 + 9x^2 + 20x}{x^2 + x - 12} = 11
\]

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Would you like more details or have any questions?

Here are some related questions to consider:
1. How does L'Hôpital's Rule help solve indeterminate limits?
2. What are the conditions for canceling factors in rational expressions?
3. What are the alternative methods for solving limits when direct substitution results in an indeterminate form?
4. How do you factor higher-degree polynomials systematically?
5. Can limits always be solved using L'Hôpital's Rule, or are there exceptions?

**Tip:** Always check for common factors between the numerator and denominator before applying more advanced methods like L'Hôpital's Rule.

lim (x -> 3) [(x^3 + 9x^2 + 20x) / (x^2 + x - 12)]

Learn how to solve the limit problem lim (x -> 3) [(x^3 + 9x^2 + 20x) / (x^2 + x - 12)]. This solution involves factoring, simplifying rational expressions, and applying L'Hôpital's Rule for indeterminate forms.

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