How do you find the vertical and horizontal asymptote and holes for #f(x)= x/(x-5)#?

1 Answer
Apr 10, 2018

Vertical Asymptote is x = 5
Horizontal Asymptote is y = 1
Hole in the Graph is none


A Hole and an Asymptote happens when any number makes the bottom equal to zero. This is because you cannot divide by zero. In the case of the problem above, 5 would make the bottom zero.

The difference between an Hole and an Asypmtote is that from from the original you can reduce (cancel) the top and bottom. Holes will cancel and Asymptotes will not. Since x does not cancel with (x-5) then x = 5 is vertical asymptote. There is no hole.

Horizontal Asymptotes depend on the term with the highest exponent both on top and bottom. In comparison, if the top exponent is higher then there is not one. If the top and bottom have the same exponent, then it is the fraction of the two numbers. If the top is smaller, then the asymptote will be y = 0. In the case above it is #x/x#, so the exponents are the same. This means the Horizontal Asymptote is y = 1 ( because #(1x)/(1x)# means the fraction is #1/1#).