Holes, vertical, horizontal, and oblique Asymptotes

By: Jonathan Joyner, Jae'Von Jones

Finding asymptotes and holes are pretty easy, you just need to know how to factor and understand a rational function.

Start with knowing what you are looking for:

A vertical asymptote is x=h or a line that the graph approaches vertically but never touches. Finding a vertical asymptote from an equation is pretty simple.  If nothing cancels out and it is left over in the denominator, it is your vertical asymptote.

For instance this is a graph with a vertical asymptote of x=-5, pretty easy to find from the graph, but lets do one from the equation.  A simple equation could be 1/(x+5)  nothing cancels out so what is left over is (x+5), simplified it is x=-5

Here is little more complicated example.

    x2 – 5x – 6 = 0
    (x – 6)(x + 1) = 0
    x = 6 or –1

So x cannot be 6 or –1, because then you would be dividing by zero.

Pretty simple right?

A hole is a single point at which the function has no value.  Finding a hole in an equation is pretty simple as well.  If it cancels out in the numerator and Denominator, it is a hole. Say if you had the equation

Then x+4 would cancel out since it is in the numerator and the denominator, therefor x=-4 is a hole.

Now on to the complicated stuff.

A horizontal asymptote is y=k or a line that the graph approaches horizontally but never touches.  There are 2 rules to Horizontal Asymptotes.

A oblique asymptote is an asymptote with a slant, or a line with a slope, to find this you must always use long division.

The Rules Are:

EXAMPLE for oblique asymptote

Once simplified it is y= 2x-2+2/x+1

The oblique asymptote is y=2x-2

Comment Stream