Question

How do you find the asymptotes for `y= (x + 1 )/ (2x - 4)`?

Answer 1

There is a vertical asymptote at `x=2`.
There is a horizontal asymptote at `y=1/2`.

Explanation:

First step is always to simplify what you have.

`frac{x+1}{2x-4} -= 1/2(1+frac{3}{x-2})`

As you know, dividing by a very large number results in almost zero. Therefore,

`lim_{x->-oo} frac{x+1}{2x-4} = lim_{x->-oo} 1/2(1+frac{3}{x-2})`
`= 1/2(1+0) = 1/2`

Similarly,

`lim_{x->oo} frac{x+1}{2x-4} = 1/2`.

Hence there is a horizontal asymptote at `y=1/2`.

Also note that `x!=2`, as that will result in division by zero.

There is a vertical asymptote at `x=2`.

Please refer to the graph below.

graph{(x+1)/(2x-4) [-10, 10, -5, 5]}