Asymptotes
Near an input it refuses, or far out to either side, a rational function’s graph follows a line. Which line, and how can you tell from the formula?
Where you are 3 inside
What this area is about
An asymptote is a line a graph gets as close to as you like. For a rational function \displaystyle f(x) = \frac{p(x)}{q(x)}, with p and q polynomials, there are two kinds of place to look: near an input where the denominator is 0, and far out to the left and right. The formula decides both, without plotting points. A graph may cross a horizontal or slant asymptote; it only has to settle toward the line far out.
How these fit together
Near an input. Vertical asymptotes sit at inputs where the denominator of the fraction in lowest terms is 0. There the outputs grow without bound, and the lesson tells this apart from a single missing point.
Far out. Horizontal asymptotes compare the degrees of p and q: when the degree of p is at most the degree of q, the graph levels off toward a horizontal line. Slant asymptotes handle a numerator one degree higher, where the graph follows a slanted line found by dividing.
A path through it
Vertical asymptotes can come at any point. Horizontal asymptotes come before slant ones.
Next: graphing rational functions, then the tangent graph and logistic growth, which levels off toward a horizontal asymptote.