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To solve this limit:

\[
\lim_{{x \to \infty}} \frac{4 + 2x - 6x^4}{x^3 - 2x^4}
\]

we'll analyze the behavior of the expression as \( x \to \infty \). Let's follow these steps:

1. **Identify the highest power of \( x \)** in both the numerator and denominator. Here, \( x^4 \) is the highest power in both parts.

2. **Divide each term by \( x^4 \)**, the highest power, to simplify:

   \[
   \frac{4 + 2x - 6x^4}{x^3 - 2x^4} = \frac{\frac{4}{x^4} + \frac{2}{x^3} - 6}{\frac{1}{x} - 2}
   \]

3. **Evaluate the limit as \( x \to \infty \)**. As \( x \to \infty \), terms like \( \frac{4}{x^4} \) and \( \frac{2}{x^3} \) approach zero. Thus, the expression simplifies to:

   \[
   \frac{0 + 0 - 6}{0 - 2} = \frac{-6}{-2} = 3
   \]

### Final Answer:
\[
\lim_{{x \to \infty}} \frac{4 + 2x - 6x^4}{x^3 - 2x^4} = 3
\]

Would you like further details or have any questions?

---

1. What if the highest powers in the numerator and denominator are different?
2. How do limits at infinity differ from limits as \( x \to 0 \)?
3. How can this technique be applied to limits approaching negative infinity?
4. How would this change if we included trigonometric functions in the expression?
5. What happens when polynomials of equal degree have the same leading coefficient?

**Tip:** Always factor out the highest power of \( x \) to simplify limits at infinity for rational functions.

lim (4 + 2x - 6x^4) / (x^3 - 2x^4) as x approaches infinity

Explore how to find the limit of the rational function (4 + 2x - 6x^4) / (x^3 - 2x^4) as x approaches infinity. Learn step-by-step how dividing by the highest power simplifies the calculation. Perfect for high school students tackling limits at infinity.

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