Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x)=x/(x-1)^2`?

Answer 1

The vertical asymptote is `x=1`
No slant asymptote.
The horizontal asymptote is `y=0`

Explanation:

The domain of `f(x)` is `D_f(x)=RR-{1}`

As you cannot divide by `0`, `x!=1`

The vertical asymptote is `x=1`

The degree of the numerator is `<` than the degree of the denominator, there is no slant asymptote.

To calculate the limits as `x->+-oo`, we take the termsof highest degree in the numerator and the denominator

`lim_(x->-oo)f(x)=lim_(x->-oo)x/x^2=lim_(x->-oo)1/x=0^(-)`

`lim_(x->+oo)f(x)=lim_(x->+oo)x/x^2=lim_(x->+oo)1/x=0^(+)`

The horizontal asymptote is `y=0`

graph{x/(x-1)^2 [-10, 10, -5, 5]}