Asymptotes

1.2 Asymptotes

We say a line is an asymptote of a curve if the distance between the line and the curve approaches zero as the curve (specifically the \(x\) or \(y\) coordinates of the points on the curve) goes to \(+\infty \) or \(-\infty \).

We study three types of asymptotes: (1) vertical, (2) horizontal, and (3) oblique (or inclined or slant).

Vertical Asymptotes

The line \(x=a\) is a vertical asymptote of the graph of \(f\) if \(f(x)\to +\infty \) or \(f(x)\to -\infty \) as \(x\) approaches \(a\) from the left or right.

See Figure 1.2.1.

PIC

Figure 1.2.1:The vertical line \(x=a\) is a vertical asymptote of a curve if the \(y\) coordinates of the points on the curve approach \(+\infty \) or \(-\infty \) as \(x\) approaches \(a\) (from the left or right or both directions).