Let's solve the problem of finding the horizontal and vertical asymptotes for the given rational function:

$r(x) = \frac{7x^2 + 6x}{x^4 - 1}$

Step 1: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided that the numerator is not zero at those points.

Set the denominator equal to zero:

$x^4 - 1 = 0$

This equation can be factored as:

$(x^2 - 1)(x^2 + 1) = 0$

Further factor $x^2 - 1$:

$(x - 1)(x + 1)(x^2 + 1) = 0$

Thus, the solutions are:

$x - 1 = 0 \quad \Rightarrow \quad x = 1$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$

$x^2 + 1 = 0$ gives:

$x^2 = -1 \quad \Rightarrow \quad x = \pm i \quad (\text{complex roots, so ignore})$

Therefore, the vertical asymptotes are at $x = 1$ and $x = -1$.

Step 2: Find Horizontal Asymptotes

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

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

For $r(x) = \frac{7x^2 + 6x}{x^4 - 1}$:

• The degree of the numerator $7x^2 + 6x$ is 2.
• The degree of the denominator $x^4 - 1$ is 4.

Since the degree of the numerator (2) is less than the degree of the denominator (4), the horizontal asymptote is:

$y = 0$

Final Answer:

• Vertical Asymptotes: $x = 1$, $x = -1$
• Horizontal Asymptote: $y = 0$

Would you like more details on any of these steps or have any further questions?

Here are some related questions:

1. How would the horizontal asymptote change if the degrees of the numerator and denominator were equal?
2. Can a function have more than one horizontal asymptote? Why or why not?
3. What happens to the function $r(x)$ as $x$ approaches infinity?
4. How do complex roots affect the location of vertical asymptotes?
5. Can there be a situation where the function has no asymptotes at all?

Tip: When finding vertical asymptotes, always check for any common factors between the numerator and denominator that might cancel out. This could lead to holes in the graph rather than asymptotes.