An asymptote of a function is a line that the function's graph approaches as the distance to the origin approaches infinity. As an example, the rational function $$f(x)=\frac{1}{x}$$ has two asymptotes, the $x$-axis and the $y$-axis.

As $x\to \infty$ and as $x\to \text{-}\infty ,$ the function value approaches $0.$ Thus, the $x$-axis, $y=0,$ is a horizontal asymptote of $f.$ Similarly, as $x\to 0,$ the function value approaches either positive or negative infinity. This means that the $y$-axis, $x=0,$ is a vertical asymptote of $f.$