Question

How do you find the vertical, horizontal or slant asymptotes for `f(x)= (3x-5)/(x-6)`?

Answer 1

vertical asymptote x = 6
horizontal asymptote y = 3

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

`((3x)/x - 5/x)/(x/x - 6/x) = (3 - 5/x)/(1 - 6/x)`

as `x to +- oo , 6/x" and " 5/x to 0 " and " y to 3/1`

`rArr y = 3 " 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{(3x-5)/(x-6) [-20, 20, -10, 10]}