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To solve the limit problem:

\[
\lim_{x \to 0} \frac{x + 3}{x^2(x - 4)}
\]

We will analyze the behavior of the expression as \( x \to 0 \).

### Step 1: Analyze the numerator and denominator at \( x = 0 \)
- The **numerator** \( x + 3 \) becomes \( 3 \) as \( x \to 0 \).
- The **denominator** is \( x^2(x - 4) \). At \( x = 0 \):
  \[
  x^2(0 - 4) = 0^2 \cdot (-4) = 0
  \]
  So, the denominator approaches 0 as \( x \to 0 \).

### Step 2: Behavior around 0
To understand the limit's behavior, we need to check the direction from both sides of 0:

#### 1. **As \( x \to 0^+ \) (from the right):**
- Numerator: \( x + 3 \to 3 \) (positive).
- Denominator: \( x^2 \) is positive, and \( (x - 4) \to -4 \), so the denominator is negative (positive \( x^2 \) times negative \( (x - 4) \)).
  \[
  \text{The fraction } \frac{3}{0^-} \text{ (positive numerator and negative denominator)} \to -\infty
  \]

#### 2. **As \( x \to 0^- \) (from the left):**
- Numerator: \( x + 3 \to 3 \) (positive).
- Denominator: \( x^2 \) is still positive (because squaring a negative number gives a positive value), and \( (x - 4) \to -4 \), so the denominator is negative.
  \[
  \text{The fraction } \frac{3}{0^-} \to -\infty
  \]

### Step 3: Conclusion
The limit approaches \( -\infty \) from both sides as \( x \to 0 \).

### Final Answer:
The limit is \( -\infty \).

Do you have any questions or want further details?

Here are 5 questions to consider:
1. What is the definition of an infinite limit?
2. How does the sign of the denominator affect the limit as \( x \to 0 \)?
3. Why is squaring a negative number always positive?
4. What is the behavior of functions involving \( x^2 \) as \( x \to 0 \)?
5. How can limits help in analyzing discontinuities?

**Tip:** Always check the sign of both the numerator and denominator when evaluating limits approaching 0 to determine the direction of infinity.

Determine the infinite limit lim (x -> 0) (x + 3) / (x^2 (x - 4)).

Learn how to determine the infinite limit lim (x -> 0) (x + 3) / (x^2(x - 4)) with detailed steps. This problem involves understanding the behavior of functions near asymptotes, crucial for calculus students. Suitable for Grades 11-12.

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