Evaluate the following expression.limx→∞ 10x3+x2−5/8−4x−4x3

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.

Evaluate the following expression. limx→∞ 10x^3 + x^2 − 5 / 8 − 4x − 4x^3

Discover how to evaluate the limit of a rational function as x approaches infinity with this step-by-step solution. Learn key concepts such as simplifying by dividing by the highest power of x and focusing on polynomial degree comparisons. Suitable for undergraduate mathematics students.

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