# Asymptote

Asymptote definition states that it is a line for a given curve such that curve approaches infinity, the distance between the curve and line tends to zero. In other words, an it can be defined as a line which is normally tangent (touching) and the curve at infinity. Asymptote (line) never touches the curve.

There are three types of asymptotes: Horizontal, Vertical and Oblique. For function x = f (y), horizontal asymptotes are horizontal lines, these are obtained when function approaches zero as 'y' tends to +∞ or −∞. Vertical asymptotes are vertical lines and oblique is a linear asymptote. When linear asymptote is parallel neither to 'x' nor to y- axis then it will be referred to oblique asymptote. Oblique asymptotes are also known as slant asymptotes.

Let’s consider the following function f(x) = x + 1 / x’ to define asymptote and plot its graph. Here in above graph line y = x and y - axis are both asymptotes.

Function f (x) is asymptotic to straight line y = m x + c (if m ≠ 0).

If Lim (x→ + ∞) [f (x) - (m x + c)] = 0,

Or

Lim (x→ - ∞) [f (x) - (m x + c)] = 0,

Among these two equations, equation y = m x + c is an oblique asymptote of ƒ(x) when 'x' tends to +∞, and in second equation, line y = m x + c is an oblique asymptote of ƒ (x) when 'x' tends to −∞.

Graph of expression A = f (B) consists of horizontal asymptote on horizontal lines. Horizontal asymptotes are obtained when expression approaches zero as 'B' tends either +∞ or −∞.

Vertical asymptotes are vertical lines near graph and in case of vertical asymptotes, function increases without any bounds.

## Asymptote Rules

Asymptote is a straight line which passes very close to curve till infinite but never intersects the curve. Asymptote is mainly of three types, (1) Horizontal Asymptotes (2) Vertical asymptotes (3) oblique asymptotes. Suppose p = f(q) is a function so any line p = a, is called its horizontal asymptotes only when following condition is true:- Limit q -> +- ∞ f(q) = a.

## Asymptote Equation

Concept of Asymptotes was given by Apolloniuse of Perga. Asymptote is a line which is very close to the curve but never intersects the curve. In simple terms we can say that asymptote is a simple line, this line never intersect the given curve. There are three types of asymptotes which are given below with equation for asymptotes:-