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Horizontal Asymptote

Definition

Let LL be a real number. The line y=Ly=L is called a horizontal asymptote of the curve y=f(x)y=f(x) if either

lim⁡x→+∞f(x)=Lorlim⁡x→−∞f(x)=L.\lim_{x\to +\infty}f(x)=L\quad \text{or}\quad \lim_{x\to -\infty}f(x)=L.

Interpretation

A horizontal asymptote describes the long-term behaviour of a function as the input grows large in magnitude.

It indicates that the function approaches a constant value.

The graph of 1/(x - 1) + 1 approaching the horizontal line y = 1 far to the left and far to the right.

The figure shows f(x)=1x−1+1f(x)=\frac{1}{x-1}+1. Here f(x)→1f(x)\to1 as x→+∞x\to+\infty and as x→−∞x\to-\infty: the line y=1y=1 is a horizontal asymptote.

A limit at infinity describes how the outputs behave far along the input axis. In the graph, this says that the vertical distance from the curve to the horizontal line tends to zero, without requiring the curve to stay on one side.

Key Results

  • A function may have different horizontal asymptotes as x→+∞x\to+\infty and x→−∞.x\to-\infty\text{.}
  • Horizontal asymptotes may or may not be attained by the function.

Examples

  • The function f(x)=1/xf(x)=1/x has horizontal asymptote y=0.y=0\text{.}
  • The function f(x)=xx2+1f(x)=\dfrac{x}{\sqrt{x^2+1}} has two different horizontal asymptotes: y=1y=1 as x→+∞x\to+\infty and y=−1y=-1 as x→−∞.x\to-\infty\text{.} The arctangent arctan⁡x\arctan x, from a later lecture, behaves in the same way with y=±π2.y=\pm\dfrac{\pi}{2}\text{.}

Prerequisites

Sources

Symbols: infinity, pi, square root

Last updated: 27 October 2026 11:52