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

1 Answer
Mar 27, 2016

Answer:

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]}