Horizontal and Vertical Asymptotes

An asymptote is a line that a graph approaches more and more closely but never actually reaches. The graph gets arbitrarily close — within any distance you can name — but no point of the curve ever lies exactly on the asymptote.

Asymptotes come in two main flavors: horizontal and vertical.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph approaches as \(x\) heads toward positive or negative infinity. Consider \(y = \frac{1}{x}\):

Graph of y = 1/x showing horizontal asymptote at y = 0

As \(x\) gets larger and larger, the numerator stays fixed at 1 but the denominator grows without bound, so the whole fraction shrinks toward zero. The values pass through \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{10}\), \(\frac{1}{10{,}000}\), and keep getting smaller — but never actually reach zero. The line \(y = 0\) is a horizontal asymptote.

The same thing happens with \(y = \frac{4x + 2}{x^2 + 1}\):

Graph of y = (4x+2)/(x²+1) approaching y = 0

The numerator grows linearly with \(x\), but the denominator grows quadratically. The denominator wins, so the fraction shrinks toward zero. Another horizontal asymptote at \(y = 0\).

Not every horizontal asymptote sits at \(y = 0\). Consider \(y = \frac{3x}{x + 2}\):

Graph of y = 3x/(x+2) with horizontal asymptote at y = 3

When \(x\) is very large, the \(+2\) in the denominator becomes negligible — the difference between dividing by 100,000 versus 100,002 is barely noticeable. So for large \(x\), the function behaves like \(\frac{3x}{x} = 3\). The horizontal asymptote is \(y = 3\).

That informal reasoning is really a limit calculation: \(\lim_{x \to \infty} \frac{3x}{x + 2} = 3\). To find a horizontal asymptote in general, look at what the function does as \(x \to \pm\infty\). The full rules for rational functions are in Finding Horizontal Asymptotes.

Vertical Asymptotes

A vertical asymptote is a vertical line the graph approaches as \(x\) gets close to a specific value. These almost always show up where the denominator of a fraction goes to zero while the numerator stays nonzero. Take \(y = \frac{4}{x - 2}\):

Graph of y = 4/(x-2) with vertical asymptote at x = 2

As \(x\) approaches 2 from the left, the numerator stays at 4 while the denominator gets very small and negative, so the whole fraction plunges toward \(-\infty\). At exactly \(x = 2\), the function is undefined — you can't divide by zero. Just to the right of \(x = 2\), the denominator is very small and positive, sending the function up toward \(+\infty\).

To find vertical asymptotes, look for values that make the denominator zero. Those are the candidates. (One caveat: if a factor cancels from both numerator and denominator, that's a hole in the graph rather than an asymptote. See Finding Asymptotes for the full procedure.)

A Function with Many Vertical Asymptotes

Trigonometric functions like secant give you vertical asymptotes at every place where the underlying cosine is zero:

Graph of y = sec(x) showing repeated vertical asymptotes

Since \(\sec(x) = \frac{1}{\cos(x)}\) and cosine equals zero at \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\), the secant function has a vertical asymptote at each of those points.

A Note on Graphing Software

Graphing calculators and computer plots sometimes connect points across an asymptote, drawing what looks like a continuous black line where there should be a break. The software is trying to connect adjacent plotted points, but it doesn't always recognize that the function jumps from \(-\infty\) to \(+\infty\). When you see a sudden vertical line in a computer-drawn graph of a rational or trigonometric function, check whether it's really part of the curve or an asymptote that got drawn in.

For the step-by-step procedure for finding asymptotes algebraically, see Finding Asymptotes and Finding Horizontal Asymptotes.

Related Topics