# Asymptote

A straight line to which a given curve eventually gets as close as we like, and stays close, is called an asymptote to the curve.

For example:

• For the graph of $y=\frac{1}{x}$, the $x$-axis and $y$-axis are both asymptotes: the curve gets as close as we like to the $x$-axis as $x$ tends towards $\infty$ or $-\infty$, and as close to the $y$-axis as we like as $x$ tends to zero.

• For the graph of $y=\ln(x)$ (where $x>0$), the $y$-axis is an asymptote: the curve gets as close as we like to it as $x$ tends towards zero.

• For the graph of $y=2$, the line $y=2$ is an asymptote: the curve $y=2$ (actually a straight line) is as close as we like to itself as $x$ tends towards $\infty$.

• For the graph of $y=\dfrac{\sin x}{x}$, the $x$-axis is an asymptote: as $x$ tends towards $\infty$ or $-\infty$, even though the graph crosses the $x$-axis infinitely often, the curve gets as close as we like to the $x$-axis and stays close.