Question

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

Answer 1

vertical asymptote x = 3
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 - 3 = 0 → x = 3 is the asymptote

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

divide terms on numerator/denominator by x

`(x/x +3/x)/(x/x-3/x)=(1+3/x)/(1-3/x)`

as `x to +- oo , y to (1+0)/(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+3)/(x-3) [-10, 10, -5, 5]}