Asymptotes

  • In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

  • In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.

  • The word asymptote is derived from the Greek means "not falling together".

  • There are three kinds of asymptotes:

    • horizontal

      • For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.

    • vertical

      • Vertical asymptotes are vertical lines near which the function grows without bound.

    • oblique

      • An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.

  • There are curve asymptotes believe it or not. A curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.

  • Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.

  • The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.

Essential Questions

How are limits related to asymptotes?