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The image shows a limit problem that asks to evaluate:

\[
\lim_{{x \to \infty}} \frac{2x^2 + 7x - 1}{4x^2 + 5}
\]

You are asked to determine what the result implies about the asymptotes of the function. Let's solve the limit first.

### Step 1: Analyze the Degrees of the Polynomials
The function is a rational function of the form:

\[
f(x) = \frac{2x^2 + 7x - 1}{4x^2 + 5}
\]

- The degree of the numerator is 2 (because of the term \(2x^2\)).
- The degree of the denominator is also 2 (because of the term \(4x^2\)).

### Step 2: Apply Limit for Large \(x\)
For limits as \(x \to \infty\), we focus on the highest-degree terms in both the numerator and the denominator.

\[
\lim_{{x \to \infty}} \frac{2x^2 + 7x - 1}{4x^2 + 5} = \lim_{{x \to \infty}} \frac{2x^2}{4x^2} = \frac{2}{4} = \frac{1}{2}
\]

### Step 3: Conclusion on Asymptotes
When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients (which is \(\frac{2}{4} = \frac{1}{2}\)).

Therefore, the function has a **horizontal asymptote** at \(y = \frac{1}{2}\), and there are **no vertical asymptotes**.

### Correct Answer:
The correct choice is **(c)**: *Hay asíntota horizontal pero no vertical*.

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Do you want more details on this solution, or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you find vertical asymptotes in rational functions?
2. What happens when the degree of the numerator is higher than the degree of the denominator?
3. What is the significance of horizontal asymptotes for the behavior of a function as \(x \to \infty\)?
4. How would you find slant asymptotes for a rational function?
5. What changes in asymptote analysis if the limit is taken as \(x \to -\infty\)?

**Tip**: When comparing the degrees of polynomials in limits, always check if they are equal or if one dominates the other for large \(x\). This helps you quickly determine horizontal or slant asymptotes.

El resultado al calcular lim(x→∞) (2x^2 + 7x - 1) / (4x^2 + 5) nos indica que: Seleccione una: a) Hay una asíntota vertical, b) Hay una asíntota horizontal, c) Hay asíntota horizontal pero no vertical, d) No hay asíntota vertical ni horizontal.

This problem explores how to calculate the limit of a rational function as x approaches infinity and identify horizontal and vertical asymptotes. The key concepts include the degrees of polynomials and their impact on the function's behavior at infinity.

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