Question

How do you find the vertical, horizontal or slant asymptotes for `y = x/(x-6)`?

Answer 1

vertical asymptote x = 6
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 - 6 = 0 → x = 6 , is the asymptote

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

divide terms on numerator/denominator by x

`rArr (x/x)/(x/x - 6/x) = 1/(1 - 6/x)`

as `x to +- oo , 6/x to 0" and " y to 1/(1-0)`

`rArr y = 1 " is the asymptote "`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{x/(x-6) [-20, 20, -10, 10]}