Certainly! Let's solve the limit without using L'Hôpital's rule by focusing on the dominant terms and simplifying the expression directly.

The limit to solve is:

\[
\lim_{{x \to \infty}} \left( \frac{\sqrt[3]{2x^2 - x^3}}{3x^2 - 2\sqrt{3x^2} - 5} \right)
\]

### Step-by-Step Solution:

1. **Simplify the Numerator and Denominator by Factoring:**

   We start by identifying the highest power of \( x \) in both the numerator and the denominator.

   In the numerator, \( 2x^2 - x^3 \) can be written as:
   \[
   \sqrt[3]{2x^2 - x^3} = \sqrt[3]{-x^3 (1 - \frac{2x^2}{x^3})} = \sqrt[3]{-x^3 (1 - \frac{2}{x})}
   \]

   For large \( x \), \( \frac{2}{x} \) approaches 0, so:
   \[
   \sqrt[3]{2x^2 - x^3} \approx \sqrt[3]{-x^3} = -x
   \]

   In the denominator, \( 3x^2 - 2\sqrt{3x^2} - 5 \) can be simplified by dividing each term by \( x^2 \):
   \[
   3x^2 - 2\sqrt{3x^2} - 5 = 3x^2 - 2\sqrt{3}x - 5
   \]
   The dominant term here is \( 3x^2 \).

2. **Rewrite the Expression:**

   Given the dominant terms, we rewrite the original limit expression as:
   \[
   \frac{\sqrt[3]{2x^2 - x^3}}{3x^2 - 2\sqrt{3x^2} - 5} \approx \frac{-x}{3x^2}
   \]

3. **Simplify the Fraction:**

   Simplify the fraction:
   \[
   \frac{-x}{3x^2} = \frac{-1}{3x}
   \]

4. **Evaluate the Limit:**

   As \( x \to \infty \):
   \[
   \lim_{{x \to \infty}} \frac{-1}{3x} = 0
   \]

### Conclusion:
\[
\lim_{{x \to \infty}} \left( \frac{\sqrt[3]{2x^2 - x^3}}{3x^2 - 2\sqrt{3x^2} - 5} \right) = 0
\]

Would you like any additional details or have any questions about this solution?

Here are 5 questions you might ask next:
1. How can we determine the dominant terms in a function as \( x \to \infty \)?
2. What techniques can be used to simplify complex fractions?
3. How do different powers of \( x \) affect the behavior of a function at infinity?
4. Can you explain how to handle limits involving different types of roots?
5. What are the key differences between using algebraic simplifications and L'Hôpital's rule for limits?

**Tip:** When dealing with limits, especially as \( x \to \infty \), focusing on the highest power terms in both the numerator and denominator can often simplify the problem and lead you to the correct answer efficiently.

Explore how to solve the limit without using L'Hôpital's rule with a detailed step-by-step guide. Learn the techniques to simplify complex expressions involving limits as x approaches infinity. Suitable for undergraduate-level mathematics.

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