Question

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

Answer 1

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`