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How to answer this? The lesson is Infinite Limit

The problem involves evaluating the **one-sided limit** as \(x \to -4^+\) of the rational function:

\[
\lim_{x \to -4^+} \frac{x^3 + 2x^2 - 3x}{x^2 - 5x + 4}.
\]

### Steps to Solve (Infinite Limit Concept):
1. **Understand the Behavior of the Denominator**:
   The denominator is:
   \[
   x^2 - 5x + 4 = (x - 4)(x - 1).
   \]
   As \(x \to -4^+\), the factor \((x - 4)\) in the denominator will approach a negative value, and \((x - 1)\) will approach \((-4 - 1 = -5)\), which is also negative.

   Therefore:
   \[
   x^2 - 5x + 4 \to \text{positive} \times \text{negative} = \text{positive}.
   \]

2. **Understand the Behavior of the Numerator**:
   The numerator is:
   \[
   x^3 + 2x^2 - 3x.
   \]
   Substitute \(x = -4\) directly to check the leading sign:
   \[
   (-4)^3 + 2(-4)^2 - 3(-4) = -64 + 32 + 12 = -20.
   \]
   So the numerator approaches a negative value as \(x \to -4^+\).

3. **Combine Results**:
   - The numerator approaches a **negative value**.
   - The denominator approaches a **positive value**.

   Therefore, the fraction approaches:
   \[
   \frac{\text{negative}}{\text{positive}} = \text{negative}.
   \]

   The limit does not exist in a finite sense because the numerator is nonzero and the denominator approaches zero. The value of the fraction grows infinitely in magnitude, but the sign is **negative**.

4. **Conclusion**:
   \[
   \lim_{x \to -4^+} \frac{x^3 + 2x^2 - 3x}{x^2 - 5x + 4} = -\infty.
   \]

Would you like additional clarification on any of the steps, or more examples?  

### 5 Related Questions to Explore:
1. What happens to this limit as \(x \to -4^-\) instead?
2. How can you determine if the limit approaches infinity or negative infinity for other similar rational functions?
3. What role does factoring the denominator and numerator play in solving infinite limits?
4. How can you apply L'Hôpital's Rule to limits involving infinity?
5. What is the geometric interpretation of limits approaching infinity?

### Tip:
For rational functions, always factor both the numerator and denominator to clearly identify the behavior near the point of interest!

Evaluate the one-sided limit lim(x→-4⁺) (x³ + 2x² - 3x) / (x² - 5x + 4).

Learn how to evaluate the one-sided limit lim(x→-4⁺) (x³ + 2x² - 3x) / (x² - 5x + 4). This solution demonstrates the concept of infinite limits, including numerator and denominator behavior and factorization techniques. Suitable for advanced high school students studying calculus.

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