Question

How do I find the vertical and horizontal asymptotes of `f(x)=x^2/(x-2)^2`?

Answer 1

• Vertical asymptotes :

These are the illegal values for x
You mustn't divide by 0, thus `(x-2)²` mustn't be equal to 0
Which means `x-2` mustn't be equal to 0
And thus `x` mustn't be equal to 2
2 is the illegal value

So there is a vertical asymptote in `x=2`

• Horizontal asymptotes :

We are looking for the limits

1. Limit in `-prop`

`lim(x²)=+prop`
`lim(x-2)²=lim(x²-4)=+prop`
The `-4` is insignificant since we deal with really great numbers
So `limf(x)=lim((x²)/(x²))=lim(1)=1`

We thus have an horizontal asymptote in `y=1`

1. Limit in `prop`

`lim(x²)=+prop`
`lim(x-2)²=lim(x²-4)=+prop`
The `-4` is insignificant since we deak with really great numbers
So `limf(x)=lim((x²)/(x²))=lim(1)=1`

Same vertical asymptote in `y=1`