# How do you tell if it's a vertical asymptote function or a horizontal asymptote function?

Oct 12, 2015

See explanation

#### Explanation:

To see if a function has vertical asymptote you have to find values of $x$ which are not in the domain, but their surrounding is. For example if $f \left(x\right) = \frac{1}{x}$, then $x = 0$ is a vertical asymptote. To ensure that such point is an asymptote you have to calculate left and right side limits:

${\lim}_{x \to {0}^{+}} \frac{1}{x} = + \infty$

${\lim}_{x \to {0}^{-}} \frac{1}{x} = - \infty$

graph{(1/x) [-10, 10, -5, 5]}

To find if a function has horizontal (or oblique) asymptotes you have to calculate limits:

$a = {\lim}_{x \to - \infty} f \frac{x}{x}$ and $b = {\lim}_{x \to - \infty} f \left(x\right) - a x$

If both limits are finite, then function $f \left(x\right) = a x + b$ is an oblique (or horizontal for $a = 0$) asymptote for $f \left(x\right)$.
For example function $f \left(x\right) = x + \frac{1}{x}$ has an oblique asymptote $f \left(x\right) = x$

graph{(y-x-1/x)(y-x)=0 [-10, 10, -5, 5]}

Oct 14, 2015

See the explanation.

#### Explanation:

Vertical

$x = h$ is a vertical asymptote if $f \left(x\right)$ increases or decreases without bound ($f \left(x\right) \rightarrow \infty \text{ or } - \infty$) as $x$ approaches $h$ from at least one side.

Roughly: A vertical asymptote occurs if $y$ goes to $\pm$ infinity as $x$ is approaching some finite number

Horizontal

Roughly: A horizontal asymptote occurs if $y$ goes to some finite number as $x$ is goes to $\pm$ infinity.

$y = k$ is a horizontal asymptote if $f \left(x\right) \rightarrow k$ as $x$ increases or decreases without bound ($x \rightarrow \infty \text{ or } - \infty$)