A remarkable line: the asymptote

The emblematic case of the hyperbola -----------------------------------

Most people first encounter an asymptote in high school, when studying the reciprocal function. In an orthonormal coordinate system, its graph, with equation $y=\frac{1}{x}y=\backslash frac\{1\}\{x\}$, has both coordinate axes as "asymptotes."

The point M on the graph, with coordinates $(x,\frac{1}{x})(\; x,\backslash frac\{1\}\{x\})$, lies at a distance $\frac{1}{x}\backslash frac\{1\}\{x\}$ from its orthogonal projection H, with coordinates (x, 0), on the x-axis. This distance approaches 0 as x approaches infinity, without ever equaling 0. Thus, the graph "approaches" the x-axis without ever reaching it.

This idea is reflected in the etymology of the word "asymptote": formed from the privative prefix "a" and the Greek symptôsis ("meeting"), it essentially means "no meeting."