To find the asymptotes for the given rational function:

$f(x) = \frac{x^2 + x - 1}{x^2 - 1},$

we analyze the following:

1. Vertical Asymptotes:

Vertical asymptotes occur when the denominator equals zero, provided the numerator does not also equal zero at the same point. Set the denominator equal to zero:

$x^2 - 1 = 0 \implies (x - 1)(x + 1) = 0.$

So, $x = 1$ and $x = -1$ are the points where the denominator becomes zero.

Now check if the numerator is also zero at these points:

• At $x = 1$: $x^2 + x - 1 = 1^2 + 1 - 1 = 1 \neq 0$,
• At $x = -1$: $x^2 + x - 1 = (-1)^2 + (-1) - 1 = -1 \neq 0$.

Thus, there are vertical asymptotes at $x = 1$ and $x = -1$.

2. Horizontal Asymptotes:

Horizontal asymptotes depend on the degrees of the numerator and denominator:

• The degree of the numerator ($x^2 + x - 1$) is 2,
• The degree of the denominator ($x^2 - 1$) is also 2.

When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator:

$\text{Horizontal asymptote: } y = \frac{\text{Coefficient of } x^2 \text{ in numerator}}{\text{Coefficient of } x^2 \text{ in denominator}} = \frac{1}{1} = 1.$

Thus, there is a horizontal asymptote at $y = 1$.

3. Oblique Asymptotes:

Oblique asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. Since the degrees of the numerator and denominator are equal, there are no oblique asymptotes in this case.

Final Answer:

• Vertical asymptotes: $x = 1$, $x = -1$,
• Horizontal asymptote: $y = 1$.

Would you like more details or a graphical explanation of these asymptotes?

Related Questions:

1. How do we verify the behavior of $f(x)$ near its vertical asymptotes?
2. How can we confirm horizontal asymptotes graphically?
3. What happens to $f(x)$ as $x \to \pm \infty$?
4. Can we find intercepts to better understand the graph?
5. What are the key differences between vertical and horizontal asymptotes?

Tip:

Always simplify the function first to check for removable discontinuities before concluding vertical asymptotes.