Question

How do you find the vertical, horizontal or slant asymptotes for `f(x)= x/(x-5)`?

Answer 1

vertical asymptote x = 5
horizontal asymptote y = 1

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : x - 5 = 0 → x = 5 is the asymptote

Horizontal asymptotes occur as `lim_(xto+-oo) , f(x)to 0`

divide terms on numerator/denominator by x

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

as `xto+-oo,f(x)to1/(1-0)`

`rArry=1" is the asymptote"`

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