This is a limit problem where you need to evaluate the given rational function as $x \to \infty$:

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

Step-by-Step Solution:

1. Identify the highest degree of $x$:

   • In the numerator: $2x^4$ is the highest term.
   • In the denominator: $3x^4$ is the highest term.

2. Divide numerator and denominator by $x^4$: To simplify, divide each term by $x^4$, the highest power:

   $\frac{2x^4 + 3x^2}{3x^4 + 5x} = \frac{\frac{2x^4}{x^4} + \frac{3x^2}{x^4}}{\frac{3x^4}{x^4} + \frac{5x}{x^4}} = \frac{2 + \frac{3}{x^2}}{3 + \frac{5}{x^3}}.$

3. Simplify as $x \to \infty$:

   • As $x \to \infty$:
     • $\frac{3}{x^2} \to 0$,
     • $\frac{5}{x^3} \to 0$.

   The expression becomes:

   $\frac{2 + 0}{3 + 0} = \frac{2}{3}.$

Final Answer:

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

Would you like further details or assistance with a similar problem? Here are 5 related questions for practice:

1. How do you find the limit of $\frac{5x^3 + x}{x^3 + 7}$ as $x \to \infty$?
2. What happens if the degree of the numerator is higher than the degree of the denominator in a rational function limit?
3. Can you explain the significance of dominant terms when evaluating limits at infinity?
4. How does L’Hopital’s Rule apply to limits at infinity?
5. What happens if both numerator and denominator contain exponential functions?

Tip: Always simplify rational functions by dividing by the highest power of $x$ to easily identify dominant terms.