Asymptote

Tags

Algebra

Calculus

Sec/5

Cegep/1

Word count

163 words

Reading time

2 minutes

Types

Vertical (V. A.)

$x=a$ is a vertical asymptote of $f(x)$ if any of the following is true: $\underset{x\to a^{\pm }}{lim}f(x)=\pm \mathrm{\infty }$

Cannot be crossed

A function can have unlimited vertical asymptotes

A limit containing an expression of the form $\frac{c}{0}$ often has a vertical asymptote
The side $0$ is approached from affects the direction of divergence (sign of $\mathrm{\infty }$)

Horizontal (H. A.)

$y=L$ is a horizontal asymptote of $f(x)$ if $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)=L$

Can be crossed

A function can have at most two horizontal asymptotes

Evaluation

Vertical

For a fraction $\frac{P(x)}{Q(x)}$,
find all $M$ such that $Q(M)=0$.
$x=M$ is a V. A. if $\underset{x\to M^{\pm }}{lim}\frac{P(x)}{Q(x)}=\pm \mathrm{\infty }$.

Horizontal

Find the limits $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)=L$.
$y=L$ is a H. A. if $L$ exists.

Contributors

Ajitani Hifumi

Changelog

Last edited about 2 hours ago

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