Question

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

Answer 1

vertical asymptote x = 3
horizontal asymptote y = 1

Explanation:

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

solve : x - 3 = 0 → x = 3 is the asymptote

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

divide terms on numerator/denominator by x

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

as x `tooo , 3/x to 0`

`rArr y = 1/1 = 1 " is the asymptote "`

slant asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here , hence there are no slant asymptotes.

Here is the graph of the function.
graph{x/(x-3) [-10, 10, -5, 5]}