Asymptote

Types

Vertical (V. A.)

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

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 $\infty$ )

Horizontal (H. A.)

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

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 $lim_{x \to M^{\pm}} \frac{P (x)}{Q (x)} = \pm \infty$ .

Horizontal

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

Contributors

Ajitani Hifumi

Changelog

Last edited 9 days ago

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Asymptote ​

Types ​

Vertical (V. A.) ​

Horizontal (H. A.) ​

Evaluation ​

Vertical ​

Horizontal ​