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Vertical Asymptote

Definition

Let x0∈R.x_0\in\mathbb{R}\text{.} The vertical line x=x0x=x_0 is called a vertical asymptote of the curve y=f(x)y=f(x) if at least one of the following holds:

lim⁡x→x0f(x)=+∞,lim⁡x→x0+f(x)=+∞,lim⁡x→x0−f(x)=+∞,lim⁡x→x0f(x)=−∞,lim⁡x→x0+f(x)=−∞,lim⁡x→x0−f(x)=−∞.\begin{aligned} \lim_{x\to x_0}f(x)&=+\infty,\\ \lim_{x\to x_0^+}f(x)&=+\infty,\\ \lim_{x\to x_0^-}f(x)&=+\infty,\\[6pt] \lim_{x\to x_0}f(x)&=-\infty,\\ \lim_{x\to x_0^+}f(x)&=-\infty,\\ \lim_{x\to x_0^-}f(x)&=-\infty. \end{aligned}

Interpretation

A vertical asymptote describes unbounded behaviour near a finite input value: as xx approaches x0x_0 from at least one side, f(x)f(x) tends to +∞+\infty or to −∞-\infty, and the graph runs along the vertical line x=x0x=x_0.

The graph of 1/(x - 1) + 1 rising to plus infinity just right of x = 1 and falling to minus infinity just left of it, with the vertical line x = 1.

The figure shows f(x)=1x−1+1f(x)=\frac{1}{x-1}+1. As x→1+x\to1^+ the term 1x−1\frac{1}{x-1} is large and positive, so f(x)→+∞f(x)\to+\infty; as x→1−x\to1^- it is large and negative, so f(x)→−∞f(x)\to-\infty. Either side alone would make the line x=1x=1 a vertical asymptote. The same graph approaches the line y=1y=1 as x→±∞x\to\pm\infty, a horizontal asymptote.

Shrink hh and watch f(x0−h)f(x_0-h) and f(x0+h)f(x_0+h): for k≠0k\ne0 the values run off to +∞+\infty on one side and −∞-\infty on the other; k=0k=0 gives a punctured constant with no vertical asymptote. Change the sign of kk to swap the sides.

Divergence to infinity at a finite input is the analytic condition behind the vertical line. At least one relevant one-sided limit must be +∞+\infty or −∞-\infty; the two sides need not have the same sign. Unboundedness alone is not enough: 1xsin⁡1x\frac{1}{x}\sin\frac{1}{x} is unbounded near 00, but it swings between large positive and large negative values and has no infinite one-sided limit, so the definition does not apply.

For a rational function p/qp/q with no common factor, the vertical asymptotes are the lines x=x0x=x_0 at the zeros of qq. When numerator and denominator share a zero, the point may instead be a removable singularity: x2−1x−1\frac{x^2-1}{x-1} has no vertical asymptote at x=1x=1.

Key Results

  • Vertical asymptotes arise from divergence to +∞+\infty or −∞-\infty at finite points.
  • One-sided divergence to +∞+\infty or −∞-\infty is sufficient; unbounded oscillation is not.
  • Vertical asymptotes are detected using limits.

Examples

  • The function f(x)=1/xf(x)=1/x has a vertical asymptote at x=0.x=0\text{.}
  • Rational functions often exhibit vertical asymptotes at zeros of the denominator.

Prerequisites

Sources

Symbols: element of, infinity, real numbers

Last updated: 27 October 2026 11:52