# How do you find the vertical, horizontal and slant asymptotes of: y = (2 + x^4)/(x^2 − x^4) ?

Feb 14, 2017

Horizontal : $y = - 1$, in both $\leftarrow \mathmr{and} \rightarrow$ directions.
Vertical :$x = 0 \uparrow , \mathmr{and} \uparrow {x}^{2} - 1 = 0 \downarrow$. See Socratic graphs

#### Explanation:

Treating y as a function of x^2, the partial fractions are

$y = - 1 + \frac{2}{x} ^ 2 + \frac{3}{{x}^{2} - 1}$-

The asymptote y = quotient = -1 is horizontal, in both $\leftarrow \mathmr{and} \rightarrow$

directions.

The denominators $x = 0 \uparrow , \mathmr{and} \uparrow {x}^{2} - 1 = 0 \downarrow$ give vertical

asymptotes.

graph{(2+x^4)/(x^2-x^4) [-40, 40, -20, 20]}

graph{((2+x^4)/(x^2-x^4)-y)(y+1)(x-1-.003y)(x+1+.003y)(x+.001y)=0 [-1.5, 1.5, -15, 15]}

Ad hoc Scale y : x is 20 : 1 for showing asymptotes