Mathématiques

What Is an Asymptote?

A line a curve approaches without meeting. Both asymptotes of y = 1/x named, the single one belonging to y = 2ˣ, and what never touches actually promises.

Asymptote

A straight line that a curve draws ever closer to as it runs out towards infinity, without ever meeting it.

Où les élèves le rencontrent: Grade 9 or 10 mathematics, when reciprocal and exponential graphs are first sketched, and named explicitly on IGCSE Extended papers.

La réponse en bref
An asymptote is a straight line that a curve gets closer and closer to without ever reaching it. The curve y = 1/x has two: the y-axis, x = 0, and the x-axis, y = 0. The curve y = 2ˣ has one, the x-axis, approached as x becomes large and negative.

Un exemple

y = 1/x : x = 0 and y = 0 y = 2ˣ : y = 0

Put small numbers into 1/x and the outputs climb without limit: x = 0.1 gives 10, x = 0.001 gives 1000. The curve races up the y-axis and never crosses it, because x = 0 is not a value the function accepts at all. Now put in large numbers: x = 1000 gives 0.001, and no input whatever makes the output exactly zero, so the curve flattens towards the x-axis forever. y = 2ˣ does the same thing at one end only — 2⁻¹⁰ is about 0.001, small but not zero.

Two asymptotes on one familiar curve

y = 1/x is the graph the word is usually introduced on, and it has an asymptote in each direction for two different reasons. The vertical one exists because x = 0 is excluded from the domain — the rule breaks there, so the curve has a gap it approaches from both sides.

The horizontal one exists because of what division can never produce. Dividing 1 by a larger and larger number gives a smaller and smaller answer, but never zero, since zero times anything is zero and never 1. So the curve settles towards the x-axis and stays above it on the right, below it on the left.

This is why the graph is in two separate pieces rather than one. The vertical asymptote is not a line the curve is avoiding out of politeness; it marks a value the function simply does not have.

Exponential curves have exactly one

y = 2ˣ climbs steeply to the right with no ceiling at all, so there is nothing horizontal for it to approach on that side. To the left it does the opposite: 2⁻¹ = 0.5, 2⁻¹⁰ is roughly 0.001, and the values keep halving without ever arriving at zero.

That single horizontal asymptote at y = 0 is the shape's signature, and it is what a mark scheme is checking for in a sketch. A curve drawn touching down and running along the axis has drawn a different function. Add the constant, as in y = 2ˣ + 3, and the asymptote lifts with it to y = 3.

What never touches actually promises

The everyday description — gets closer forever without touching — is exactly right for every curve met at GCSE and IGCSE, and it is the right thing to have in mind when sketching. Draw the asymptote as a dashed line and keep the curve off it.

The fuller statement is slightly different, and worth knowing so the rule is not carried too far. What an asymptote guarantees is that the gap between curve and line shrinks towards nothing in the long run. A curve that oscillates while dying away can cross its horizontal asymptote repeatedly and still approach it, which happens with functions met after school. Never touching is a description of the school cases, not a law.

Questions fréquentes

What are the asymptotes of y = 1/x?

The two axes: x = 0 and y = 0. The vertical one comes from the domain, since dividing by zero is undefined, and the horizontal one from the outputs, since 1 divided by any number is never exactly zero. Sketches are expected to show the curve approaching both without meeting either.

Can a curve ever cross its asymptote?

Not the ones you meet at GCSE and IGCSE. In general the condition is only that the gap tends to zero in the long run, and some curves studied later do cross a horizontal asymptote before settling towards it. For reciprocal and exponential graphs, treat never touching as the working rule.

Should asymptotes be drawn on a sketch?

Yes, as dashed lines, and label them with their equations. Examiners look for the curve bending away from the asymptote rather than flattening onto it, and a sketch that runs along the axis is usually marked as a different curve. When the asymptote is an axis, showing the curve staying clear is enough.

Is a vertical asymptote the same as an excluded value?

Closely tied but not identical wording. The excluded value is a fact about the domain — x = 3 is not allowed in 1/(x − 3). The vertical asymptote x = 3 is what that exclusion looks like on the graph. Finding one is usually the fastest way to find the other.

Dernière mise à jour 21 août 2026