The given expression is:

$\lim_{x \to \infty} \frac{5x^3 - 2x^2 + 1}{3x + 5}$

To solve this limit as $x \to \infty$, we look at the degrees of the terms in the numerator and denominator.

Step 1: Compare the degrees of the numerator and denominator.

• The degree of the numerator $5x^3 - 2x^2 + 1$ is 3 (due to the term $5x^3$).
• The degree of the denominator $3x + 5$ is 1 (due to the term $3x$).

Step 2: Simplifying the limit.

When the degree of the numerator is greater than the degree of the denominator (3 > 1), the limit of the rational function as $x \to \infty$ approaches infinity.

Thus, the limit is: $\lim_{x \to \infty} \frac{5x^3 - 2x^2 + 1}{3x + 5} = \infty$

The result is infinity because the numerator grows much faster than the denominator as $x$ becomes large.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What if the degrees of the numerator and denominator are equal in a limit?
2. How would the limit change if the degrees of the numerator and denominator were reversed?
3. How can you apply L'Hôpital's Rule to solve indeterminate forms in limits?
4. How do limits behave for rational functions where both the numerator and denominator contain multiple terms?
5. What are some common techniques for evaluating limits as $x \to \infty$?

Tip: Always check the degrees of the polynomial terms in both the numerator and denominator when dealing with limits at infinity!