What are the asymptotes of f(x)=-x/((x-2)(4x-3) ?

1 Answer
Jun 26, 2018

Vertical Asymptote at x=2,3/4

Horizontal Asymptote at y=0

Explanation:

Vertical Asymptotes are found where a function is undefined.

The only way to make our function undefined is having 0 in the Denominator.

(x-2)(4x-3)=0 x=2,3/4

These are our Vertical Asymptotes.

To find Horizontal Asymptotes we must look at our function at -oo,oo

lim_(x->-oo)-x/((x-2)(4x-3))

Substitution gives oo/oo

Use color(blue)"L'Hopital's Rule"

lim_(x->-oo)(color(blue)(d/dx))*-x/((x-2)(4x-3))=-1/(8x-11)

Substitution give -1/-oo rarr 1/oo which goes to 0

Now look at the other side

lim_(x->oo)-x/((x-2)(4x-3))

Substitution gives -oo/oo

Use color(blue)"L'Hopital's Rule"

lim_(x->oo)(color(blue)(d/dx))*-x/((x-2)(4x-3))=-1/(8x-11)

Substitute

-1/oo which goes to 0

Limits agree :. Horizontal Asymptote at y=0