To evaluate the given limit:

$\lim_{x \to \infty} \frac{4x^2 + 50}{x^3 - 85}$

Step 1: Analyze the degrees of the numerator and denominator

• The degree of the numerator ($4x^2 + 50$) is 2.
• The degree of the denominator ($x^3 - 85$) is 3.

Since the degree of the denominator is greater than the degree of the numerator, the limit will approach 0 as $x \to \infty$.

Step 2: Formal calculation

Divide both the numerator and denominator by the highest power of $x$ in the denominator, $x^3$:

$\lim_{x \to \infty} \frac{4x^2 + 50}{x^3 - 85} = \lim_{x \to \infty} \frac{\frac{4x^2}{x^3} + \frac{50}{x^3}}{\frac{x^3}{x^3} - \frac{85}{x^3}}$

Simplify each term:

$\lim_{x \to \infty} \frac{\frac{4}{x} + \frac{50}{x^3}}{1 - \frac{85}{x^3}}$

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

• $\frac{4}{x} \to 0$
• $\frac{50}{x^3} \to 0$
• $\frac{85}{x^3} \to 0$

Thus, the expression simplifies to:

$\frac{0 + 0}{1 - 0} = 0$

Final Answer:

$\lim_{x \to \infty} \frac{4x^2 + 50}{x^3 - 85} = 0$

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Would you like a detailed explanation of dividing polynomials or how to approach similar limits?

Related Questions:

1. What happens if the numerator and denominator have the same degree in a limit?
2. How do you find limits as $x \to -\infty$?
3. What is the behavior of rational functions at their vertical asymptotes?
4. Can you explain how L'Hopital's Rule applies to limits like these?
5. What are higher-order approximations for small terms in limits?

Tip: Always compare the degrees of the numerator and denominator first to simplify limits of rational functions!