Point at infinity

A single extra point, denoted $\infty$ , added to the complex plane to describe where $z$ goes as $|z|\to\infty$ .

The point at infinity, denoted $\infty$ , is a single extra point adjoined to the complex plane to give a location to describe what happens as $|z|$ grows without bound. Unlike the real line, which has two ends ( $+\infty$ and $-\infty$ ), the complex plane gets only one point at infinity, approached the same way regardless of the direction $z$ heads off in.

Examples

Writing $\lim_{z\to\infty} 1/z = 0$ uses the point at infinity to describe $1/z$ shrinking as $|z|$ grows in any direction.
A rational function's behavior 'at infinity' is studied by substituting $z=1/w$ and examining the result as $w\to 0$ .