How to find the asymptotes of #f(x)= (x^2+4)/(6x-5x^2)#?

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Answer 1 · 2016-01-09T15:16:17

Vertical asymptotes at #x=0,6/5#, horizontal asymptote at #y=-1/5#

Vertical Asymptotes

These will occur when the denominator equals #0#.

#6x-5x^2=0#

#x(6-5x)=0#

Split this into two equations.

#x=0#

and

#6-5x=0#

#x=6/5#

The vertical asymptotes occur at #x=0,6/5#.

Horizontal Asymptotes

When the numerator and denominator have the same degree, the horizontal asymptote will be the terms with the largest degree divided.

#x^2/(-5x^2)=-1/5#

There is a horizontal asymptote at #y=-1/5#.

The function graphed:

graph{(x^2+4)/(6x-5x^2) [-10.35, 12.15, -4.37, 6.88]}