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#