Vertical Asymptote

A vertical asymptote is a vertical line, $x\=a$, that has the property that either:

1. $\underset{x\to a^{-}}{lim}f\left(x\right)\= \pm \infty$

2. $\underset{x\to a^{\+}}{lim}f\left(x\right) \= \pm \infty$

That is, as $x$ approaches $a$ from either the positive or negative side, the function approaches infinity.

Vertical asymptotes occur at the values where a rational function has a denominator of 0. The function is undefined at these points.

Horizontal Asymptote

A horizontal asymptote is a horizontal line, $y\=a$, that has the property that either:

1. $\underset{x\to \infty }{lim}f\left(x\right)\=a$

2. $\underset{x\to -\infty }{lim}f\left(x\right) \=a$

Horizontal asymptotes occur when the numerator of a rational function has degree less than or equal to the degree of the denominator. If the denominator has degree $n$, the horizontal asymptote can be calculated by dividing the coefficient of the $x^n$-th term of the numerator (it may be 0 if the numerator has a smaller degree) by the coefficient of the $x^n$-th term of the denominator.

Oblique Asymptote

An oblique or slant asymptote is an asymptote along a line $y\&equals;\mathrm{mx}\&plus;b$, where $0<\left|m\right|<\infty$. Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.