Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x) = (x^2) / (x-2)`?

Answer 1

The vertical asymptote is `x=2`
The oblique asymptote is `y=x+2`
No horizontal asymptote

Explanation:

The domain of `f(x)` is `D_f(x)=RR-{2}`

As we cannot divide by `0`, `x!=2`

Therefore,

The vertical asymptote is `x=2`

As the degree of the numerator is `>` than the degree of the denominator, we have an oblique asymptote.

Let's do a long division

`color(white)(aaaa)` `x^2` `color(white)(aaaaaaaa)` `∣` `x-2`

`color(white)(aaaa)` `x^2-2x` `color(white)(aaaa)` `∣` `x+2`

`color(white)(aaaaa)` `0+2x`

`color(white)(aaaaaaa)` `+2x-4`

`color(white)(aaaaaaaa)` `+0+4`

So,
`f(x)=x+2+(4)/(x-2)`

Now we calculate the limits

`lim_(x->-oo)(f(x)-(x+2))=lim_(x->-oo)(4)/(x-2)=0^(-)`

`lim_(x->+oo)(f(x)-(x+2))=lim_(x->+oo)(4)/(x-2)=0^(+)`

So,

The oblique asymptote is `y=x+2`

graph{(y-(x^2)/(x-2))(y-x-2)=0 [-28.86, 28.86, -14.44, 14.44]}