Asymptotes calculator: vertical, horizontal and oblique

Three kinds of asymptote, and the limit behind each

An asymptote is a straight line the graph gets arbitrarily close to as it runs off to infinity. There are three kinds, and each is found with a limit — which is why asymptotes come right after limits in every study of a function.

A vertical asymptote x = a appears where the function blows up at a finite point: at least one of the one-sided limits at a is +∞ or −∞. It is looked for at the points excluded from the domain, typically the zeros of a denominator. Not every excluded point gives one: (x² − 4)/(x − 2) tends to 4 as x → 2, and there the graph has only a hole.

A horizontal asymptote y = L appears when f(x) tends to a finite L as x → +∞ or as x → −∞. When the limit is infinite instead, there may be an oblique asymptote y = mx + q: the slope is m = lim f(x)/x, which must be finite and non-zero, and the intercept is q = lim (f(x) − mx). For a rational function this happens exactly when the numerator has degree one more than the denominator.

Common mistakes

Calling every zero of the denominator a vertical asymptote without computing the limit: if the numerator is zero there too, the point may be a removable hole.
Taking q = lim f(x) instead of q = lim (f(x) − mx). The intercept comes after the slope, and depends on it.
Assuming the asymptote as x → +∞ is also the one as x → −∞. eˣ has y = 0 only on the left, and √(x² + 1) has y = x on the right and y = −x on the left.
Believing a graph can never cross its asymptote. It can, even infinitely often: sin x / x crosses y = 0 at every multiple of π.

Frequently asked questions

How do I find the vertical asymptotes?

Take the points excluded from the domain, usually where a denominator is zero, and compute the one-sided limits there. If at least one is +∞ or −∞, the line x = a is a vertical asymptote.

How do I find an oblique asymptote?

Only where lim f(x) is infinite as x → ±∞. Compute m = lim f(x)/x: if it is finite and not zero, compute q = lim (f(x) − mx). If q is finite too, y = mx + q is the oblique asymptote.

Can a function have both a horizontal and an oblique asymptote?

Not on the same side: as x → +∞ it has one, the other or neither. On opposite sides it can, as with a function that levels off to the left and runs along a slanting line to the right.

Which rational functions have an oblique asymptote?

Those whose numerator has degree exactly one more than the denominator. Dividing the polynomials gives the quotient mx + q, which is the asymptote, and a remainder that tends to zero.

How this calculation works

Vertical asymptote x = a: lim x→a⁻ f(x) = ±∞ or lim x→a⁺ f(x) = ±∞, looked for at the finite ends of the domain. Horizontal asymptote y = L: lim x→+∞ f(x) = L or lim x→−∞ f(x) = L, with L finite. Oblique asymptote y = mx + q, as x → +∞ or as x → −∞: m = lim f(x)/x, finite and non-zero, and q = lim (f(x) − mx), finite. On one side a function has at most one horizontal or oblique asymptote. Each limit is found numerically and written exactly when recognised.

Study of a function

Domain, sign, limits, asymptotes, turning points, concavity and graph, in one go.

Domain of a function

The conditions of existence and the domain written as intervals.

Limits

One-sided and two-sided limits, at a point or at infinity, with the indeterminate form.

Derivatives

The derivative of a function and the rules used to get it.