To evaluate the expression

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

we will first analyze the degrees of the polynomials in both the numerator and denominator.

Step 1: Identify the highest degree terms

• The highest degree term in the numerator is $10x^3$.
• The highest degree term in the denominator is $-4x^3$.

Step 2: Simplify by dividing both the numerator and denominator by $x^3$ (the highest power of $x$):

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

This simplifies to: $\lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{5}{x^3}}{\frac{8}{x^3} - \frac{4}{x^2} - 4}.$

Step 3: Evaluate the limit as $x \to \infty$

As $x \to \infty$, the terms involving $\frac{1}{x}$, $\frac{1}{x^2}$, and $\frac{1}{x^3}$ will all approach 0. Thus, the expression becomes:

$\lim_{x \to \infty} \frac{10 + 0 - 0}{0 - 0 - 4} = \frac{10}{-4} = -\frac{5}{2}.$

Final Answer:

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

Would you like further details on any step?

Related Questions:

1. What happens if the leading terms of both the numerator and denominator are of the same degree?
2. How does dividing by the highest power of $x$ help in evaluating limits?
3. Can this method be applied to rational functions with lower degree terms?
4. What would happen if the denominator had a higher degree than the numerator?
5. How do limits involving infinity help in understanding asymptotic behavior?

Tip: When evaluating limits of rational functions as $x \to \infty$, focus on the highest degree terms, as lower degree terms become negligible.