To find the asymptotes of the function \( y = \frac{x^2 + x}{x^2 - 1} \), we'll consider both vertical and horizontal asymptotes.

### 1. Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is zero, provided that the numerator is not zero at those points.

The denominator is:
\[
x^2 - 1 = 0
\]

Solve for \( x \):
\[
x^2 = 1
\]
\[
x = \pm 1
\]

These values correspond to potential vertical asymptotes. We check the numerator at these points:
\[
\text{Numerator: } x^2 + x \text{ evaluated at } x = 1 \text{ and } x = -1
\]
\[
\text{At } x = 1: 1^2 + 1 = 2
\]
\[
\text{At } x = -1: (-1)^2 + (-1) = 0
\]

The numerator is zero at \( x = -1 \), so \( x = -1 \) is a removable discontinuity, not a vertical asymptote. However, at \( x = 1 \), the numerator is not zero, so \( x = 1 \) is a vertical asymptote.

**Vertical asymptote**: \( x = 1 \)

### 2. Horizontal Asymptote
To find the horizontal asymptote, we examine the behavior of the function as \( x \) approaches infinity.

The degrees of the numerator and denominator are both 2 (since the highest power of \( x \) in both the numerator and denominator is \( x^2 \)). For rational functions, if the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients.

The leading coefficient of the numerator \( x^2 + x \) is 1, and the leading coefficient of the denominator \( x^2 - 1 \) is also 1. Therefore, the horizontal asymptote is:

\[
y = \frac{1}{1} = 1
\]

**Horizontal asymptote**: \( y = 1 \)

### Summary
- **Vertical asymptote**: \( x = 1 \)
- **Horizontal asymptote**: \( y = 1 \)

Would you like more details on this process or have any questions?

Here are 5 related questions:
1. What is the behavior of the function near the vertical asymptote \( x = 1 \)?
2. How do you determine the presence of slant asymptotes?
3. What happens to the function as \( x \) approaches the removable discontinuity \( x = -1 \)?
4. Can the function \( y = \frac{x^2 + x}{x^2 - 1} \) have oblique asymptotes?
5. How would the asymptotes change if the numerator had a higher degree than the denominator?

**Tip:** When analyzing asymptotes, always start by factoring the numerator and denominator to check for common factors (removable discontinuities).

y = (x^2 + x)/(x^2 - 1) find the asymptote

Explore how to find vertical and horizontal asymptotes for the function y = (x^2 + x)/(x^2 - 1). This detailed guide covers key algebraic concepts like rational functions, asymptotes, and removable discontinuities, perfect for students in Grades 10-12.

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