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

Answer 1

Cameron Alexander

Vertical Asymptote at #x=2,3/4#

Horizontal Asymptote at #y=0#

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#

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Answer 2

Cameron Alexander

The function f(x)=-x/((x-2)(4x-3) has two vertical asymptotes at x=2 and x=3/4. There are no horizontal asymptotes.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.