Question

1. - / 4 points find the horizontal and vertical asymptotes of the curve. $y = \frac{7 + x^{4}}{x^{2}-x^{4}}$ $x=$ (smallest x - value) $x=$ $x=$ (largest x - value) $y=$

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to zero: $x^{2}-x^{4}=x^{2}(1 - x^{2})=x^{2}(1 - x)(1 + x)=0$.

Step2: Solve for x

$x^{2}=0$ gives $x = 0$, $1 - x=0$ gives $x = 1$, and $1 + x=0$ gives $x=-1$.

Step3: Find horizontal asymptotes

Divide numerator and denominator by $x^{4}$: $y=\frac{\frac{7}{x^{4}}+1}{\frac{1}{x^{2}} - 1}$. As $x\to\pm\infty$, $\frac{7}{x^{4}}\to0$ and $\frac{1}{x^{2}}\to0$. So $y=-1$.

Answer:

$x=-1$
$x = 0$
$x = 1$
$y=-1$