Vertical Asymptotes and Discontinuities Examples

Find the vertical asymptotes and holes of .

Step one, we want to factor everything as much as possible.

We know, just looking at the denominator, that there will be a hole or a vertical asymptote at x = 1 and 4. They could both be holes, or they could both be asymptotes, or we could have one of each. We have to simplify the expression to find out.

So, we have a vertical asymptote at x = 1, and a hole at x = 4.

Find the vertical asymptotes and holes of .

Hey, is this even a rational function? We demand our money back. Well, except that 15 is a polynomial, with the only term being 15x⁰.

To find holes and vertical asymptotes, we need to find all the places where the denominator equals 0. Where does 15 = 0? Only in our worst nightmares. Like the one where we show up to school naked, unprepared for a math test, and then the clowns on flaming motorcycles burst through the windows and conscript us into their gang.

Anyway, what this means is that there are no vertical asymptotes or holes. Also, we shouldn't have so much caffeine right before bedtime.

Find the vertical asymptotes and holes of .

Like any great action hero, we factor first, ask questions later.

y = x – 2

Our rational expression looks like a regular line. However, we can't forget the discontinuity from the original expression. There is a hole at x = 3.