When sketching curves, asymptotes plays a very important role as it determines the behavior of the curve at infinity.

Horizontal asymptotes

$y=a$ where $a \in \mathbb{R}$ is a horizontal asymptote to a curve $y=f(x)$ if $y \to a$ as $x \to \infty$ and/or $y \to a$ as $x \to -\infty.$

Another way to represent the result is to say that ${\displaystyle \lim_{x \to \infty} f(x) = a.}$

Vertical asymptotes

$x=b$ where $b \in \mathbb{R}$ is a vertical asymptote to a curve $y=f(x)$ if $y \to \infty$ as $x \to b.$*

* For technical reasons, the more correct definition is if $y \to \infty$ as $x \to b^+$ and/or $y \to -\infty$ as $x \to b^+$ and/or $y \to \infty$ as $x \to b^-$ and/or $y \to -\infty$ as $x \to b^-$.

The representation in limit notation for the first case is ${\displaystyle \lim_{x \to b^+} f(x) = \infty.}$

Rectangular hyperbolas

The graph of $\displaystyle y=a+\frac{c}{x-b}$ has asymptotes $y=a$ and $x=b.$

Examples