How do you find the vertical, horizontal and slant asymptotes of: y = (2 + x^4)/(x^2 − x^4) y=2+x4x2−x4?

1 Answer
A. S. Adikesavan
Feb 14, 2017

Horizontal : y = -1y=−1, in both larr and rarr←and→ directions.
Vertical :x = 0 uarr, and uarr x^2-1=0 darrx=0↑⏐,and↑⏐x2−1=0⏐↓. See Socratic graphs

Explanation:

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

y=-1+2/x^2+3/(x^2-1)y=−1+2x2+3x2−1-

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

directions.

The denominators x = 0 uarr, and uarr x^2-1=0 darrx=0↑⏐,and↑⏐x2−1=0⏐↓ 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

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