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