Question

How do you find the vertical, horizontal and slant asymptotes of: `y=(2x )/ (x-5)`?

Answer 1

vertical asymptote `x = 5`
horizontal asymptote `y = 2`

Explanation:

For this rational function (fraction) the denominator cannot be zero. This would lead to division by zero which is undefined. By setting the denominator equal to zero and solving for `x` we can find the value that `x` cannot be. If the numerator is also non-zero for such a value of `x` then this must be a vertical asymptote.

solve : `x - 5 = 0 rArr x = 5` is the asymptote

Horizontal asymptotes occur as

`lim_(xto+-oo) ytoc" (a constant)"`

divide terms on numerator/denominator by `x`

`((2x)/x)/(x/x-5/x)=2/(1-5/x)`

as `xto+-oo, yto2/(1-0)`

`rArry=2" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( both degree 1 ) Hence there are no slant asymptotes.
graph{(2x)/(x-5) [-20, 20, -10, 10]}