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

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Answer 1 · 2017-02-14T16:24:03

Horizontal : #y = -1#, in both #larr and rarr# directions.
Vertical :#x = 0 uarr, and uarr x^2-1=0 darr#. See Socratic graphs

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

#y=-1+2/x^2+3/(x^2-1)#-

The asymptote y = quotient = -1 is horizontal, in both #larr and rarr#

directions.

The denominators #x = 0 uarr, and uarr x^2-1=0 darr# 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