To solve the limit without using L'Hôpital's Rule, let’s analyze the behavior of the function as $x \to 0$ for the expression:

$\lim_{{x \to 0}} \frac{4x^6 - 8x^4 + 10x^3 + 2x^2}{5x^5 + 3x^3 + 10x}$

Step-by-Step Solution:

1. Identify the Dominant Terms: Since we are approaching $x = 0$, we should focus on the terms with the smallest powers of $x$ because they will dictate the behavior of the function near zero. Let's break down both the numerator and the denominator separately:

   • Numerator: $4x^6 - 8x^4 + 10x^3 + 2x^2$
   • Denominator: $5x^5 + 3x^3 + 10x$

2. Divide by the Smallest Power in the Denominator ( $x$ ): To simplify, divide both the numerator and the denominator by $x$, which is the smallest power of $x$ in the denominator.

   $= \lim_{{x \to 0}} \frac{\frac{4x^6}{x} - \frac{8x^4}{x} + \frac{10x^3}{x} + \frac{2x^2}{x}}{\frac{5x^5}{x} + \frac{3x^3}{x} + \frac{10x}{x}}$

   Simplify each term:

   • Numerator: $\frac{4x^6}{x} = 4x^5$, $\frac{-8x^4}{x} = -8x^3$, $\frac{10x^3}{x} = 10x^2$, and $\frac{2x^2}{x} = 2x$.
   • Denominator: $\frac{5x^5}{x} = 5x^4$, $\frac{3x^3}{x} = 3x^2$, and $\frac{10x}{x} = 10$.

   Thus, we have:

   $= \lim_{{x \to 0}} \frac{4x^5 - 8x^3 + 10x^2 + 2x}{5x^4 + 3x^2 + 10}$

3. Evaluate the Limit by Substituting $x = 0$: Now, as $x \to 0$, all terms containing $x$ in the numerator and the denominator will approach zero. This leaves us with:

   $= \frac{0}{10} = 0$

Final Answer:

$\lim_{{x \to 0}} \frac{4x^6 - 8x^4 + 10x^3 + 2x^2}{5x^5 + 3x^3 + 10x} = 0$

Would you like more details on any part of this solution?

Related Questions:

1. How would the solution change if we had to use L'Hôpital's Rule?
2. What if we were approaching a different limit, such as $x \to \infty$?
3. Can we generalize this method for limits approaching zero in other rational functions?
4. How can we factor polynomials to further simplify limits without L'Hôpital's Rule?
5. What would happen if the numerator and denominator had the same lowest power of $x$?

Tip:

When evaluating limits as $x \to 0$, look for the terms with the smallest powers in the denominator, as these will often dictate the limit's behavior.