Vertical Asymptote

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

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

or

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

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

Vertical asymptotes occur at the values where a rational function has a denominator of zero. The function is undefined at these points because division by zero mathematically ill-defined. For example, the function $f\left(x\right) \= \frac{1}{x}$ has a vertical asymptote at $x\=0$.

Horizontal Asymptote

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

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

or

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

This means, that as $x$ approaches positive or negative infinity, the function tends to a constant value $a$.

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 zero if the numerator has a smaller degree) by the coefficient of the $x^n$-th term of the denominator. For example, the function $f\left(x\right) \= \frac{x^2\+1}{x^3}\+7$ goes to 7 as $x$ approaches $\pm \infty$.

Oblique Asymptote

An oblique or slant asymptote is an asymptote along a line $y\=\mathrm{mx}\+b$, where $m\ne 0$. Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.

For example, the function $f\left(x\right) \= x\+\frac{1}{x}$ has an oblique asymptote about the line $y\=x$ and a vertical asymptote at the line $x\=0$.