Maths · 3 Functions

3.5 Asymptotes

Before you start

The shapes of the reciprocal, exponential and tangent graphs (3.1).
That dividing by zero is undefined.
Writing the equation of a horizontal or vertical line, e.g. $y = 3$ or $x = - 2$ .

Syllabus objectives covered (1)

Understand the concept of asymptotes and identify simple examples parallel to the axes on a graph.

Part 1 of 3

What an asymptote is

An asymptote is a line that a curve gets closer and closer to, without ever touching or crossing it.
The word 'ever' is doing real work. However far you follow the curve, the gap keeps shrinking but never becomes zero.
In this syllabus you only need asymptotes parallel to the axes — vertical lines $x = something$ and horizontal lines $y = something$ .
The syllabus also says algebraic derivation of asymptotes is excluded. You identify them from a graph or from an obvious feature of the function; you are never asked to prove where they are.
Write an asymptote as an equation of a line, not as a number. The answer is $x = 0$ , not '0', and $y = 2$ , not '2'. A bare number is not a line.
On a sketch, draw asymptotes as dashed lines. That is the standard convention and it makes clear the line is a guide, not part of the curve.

Part 2 of 3

Where they come from

Vertical asymptotes appear where the function is undefined — almost always where a denominator would be zero.
For $f (x) = a / x$ , the function is undefined at $x = 0$ , so $x = 0$ (the $y$ -axis) is a vertical asymptote.
For $f (x) = \frac{3}{x - 2}$ , the denominator is zero at $x = 2$ , so the vertical asymptote is $x = 2$ . Set the denominator to zero and solve — that is as far as the identification goes.
As the curve approaches a vertical asymptote it shoots off towards $+ \infty$ on one side and $- \infty$ on the other.
Horizontal asymptotes appear where the curve levels off as $x$ becomes very large or very negative.
For $f (x) = a / x$ , the value shrinks towards zero as $x$ grows, so $y = 0$ (the $x$ -axis) is a horizontal asymptote.
For an exponential such as $f (x) = 2^{x}$ , the curve flattens towards the $x$ -axis as $x$ becomes large and negative but never reaches it, so $y = 0$ is again the asymptote.
Dashed lines the curve approaches but never meets.
The tangent graph has vertical asymptotes at $90 °$ , $270 °$ , and every $180 °$ after that — the syllabus notes name these explicitly.
That is because $\tan x = \sin x / \cos x$ , and $\cos x = 0$ at exactly those angles. Same reason as before: a zero denominator.

Part 3 of 3

Reading them off a graph

Given a printed graph, look for two things: where the curve shoots off vertically, and where it flattens out horizontally.
A curve shooting up or down near a particular $x$ value means a vertical asymptote there. Read the value off the $x$ -axis and write $x = \dots$
A curve flattening towards a particular height at either end means a horizontal asymptote. Read the value off the $y$ -axis and write $y = \dots$
Do not assume a horizontal asymptote is always $y = 0$ . A curve can level off towards $y = 3$ , and reading it as the $x$ -axis out of habit loses the mark.
A curve can have both kinds at once — a reciprocal-type graph normally has one of each.
One caution: a curve may cross a horizontal asymptote somewhere in the middle before settling towards it at the ends. It is the long-run behaviour that matters. A vertical asymptote is never crossed.

Worked examples

Example 1. Write down the equations of the asymptotes of the graph of $f (x) = \frac{4}{x - 3}$ .

Show the working

Look for the vertical asymptote first: find where the function is undefined by setting the denominator to zero.
$x - 3 = 0 \Rightarrow x = 3$
So there is a vertical asymptote at $x = 3$ .
Now the horizontal one: consider what happens as $x$ becomes very large. The denominator grows without limit, so the fraction shrinks towards zero.
The curve therefore levels off towards the $x$ -axis, giving a horizontal asymptote at $y = 0$ .
Write both as equations of lines, not as numbers.

Answer: $x = 3$ and $y = 0$ .

Example 2. A graph is shown of a curve that rises steeply on the left, levels off towards the height $y = 5$ on the right, and never crosses that level. State the equation of the asymptote and explain what it tells you about the curve.

Show the working

The levelling-off is horizontal, so this is a horizontal asymptote.
Read its height off the $y$ -axis: the curve settles towards $5$ , so the asymptote is the line $y = 5$ .
Write it as an equation of a line, not as the number 5.
Interpret it: as $x$ increases the curve gets closer and closer to the height 5 without ever reaching it.
Note that this is not $y = 0$ . Assuming a horizontal asymptote must be the $x$ -axis is a common and costly habit.

Answer: $y = 5$ . As $x$ increases the curve approaches the height $5$ ever more closely but never reaches or crosses it.

Example 3. State the equations of the asymptotes of $y = \tan x$ in the interval $0 ° \leq x \leq 360 °$ , and explain why they occur there.

Show the working

Recall that $\tan x$ can be written as $\frac{\sin x}{\cos x}$ .
The function is undefined wherever the denominator $\cos x$ is zero, so look for the angles where cosine vanishes.
In the interval $0 °$ to $360 °$ , $\cos x = 0$ at $x = 90 °$ and $x = 270 °$ .
So the asymptotes are the vertical lines $x = 90 °$ and $x = 270 °$ .
The tangent curve shoots off to $+ \infty$ approaching each from the left and comes up from $- \infty$ on the other side, which is exactly the behaviour of a vertical asymptote.

Answer: $x = 90 °$ and $x = 270 °$ , because $\tan x = \sin x / \cos x$ and $\cos x = 0$ at those angles.

Where people go wrong

Mistake: Giving an asymptote as a number, e.g. writing '2'.

Instead: Write the equation of a line: $x = 2$ or $y = 2$ .

An asymptote is a line. A bare number does not say whether it is vertical or horizontal.

Mistake: Assuming every horizontal asymptote is $y = 0$ .

Instead: Read the level the curve settles towards off the graph.

Many curves level off at other heights, and the $x$ -axis is only the special case.

Mistake: Drawing a curve touching or crossing a vertical asymptote.

Instead: The curve approaches it but never reaches it.

The function is undefined there, so it has no value on that line at all.

Mistake: Trying to derive asymptotes algebraically with limits.

Instead: Identify them from the graph or from a zero denominator.

The syllabus explicitly excludes algebraic derivation, so that work earns nothing.

Mistake: Drawing asymptotes as solid lines.

Instead: Draw them dashed.

A solid line reads as part of the graph. The dashes show it is a guide the curve approaches.

Check yourself

Define an asymptote.
Answer
A line that a curve approaches ever more closely but never touches or crosses.
State the asymptotes of $y = \frac{6}{x}$ .
Answer
$x = 0$ and $y = 0$ .
State the vertical asymptote of $y = \frac{1}{x + 5}$ .
Answer
$x = - 5$ , where the denominator is zero.
Why does $y = 3^{x}$ have a horizontal asymptote at $y = 0$ ?
Answer
As $x$ becomes large and negative, $3^{x}$ gets ever closer to zero without reaching it, so the curve approaches the $x$ -axis.
How should an asymptote be drawn on a sketch?
Answer
As a dashed line, so it is clearly a guide rather than part of the curve.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. The diagram shows the graph of $y = \frac{2}{x - 1} + 3$ .

(a) Write down the equation of the vertical asymptote. 1 mark No calculator

Mark scheme

B1 x = 1

(b) Write down the equation of the horizontal asymptote. 1 mark No calculator

Mark scheme

B1 y = 3

(c) Explain what is meant by saying that $x = 1$ is an asymptote of this curve. 2 marks No calculator

Mark scheme

B1 the curve gets closer and closer to the line x = 1
B1 but never touches or crosses it / the function is undefined at x = 1

Question 2. A student sketches $y = \tan x$ for $0 ° \leq x \leq 360 °$ .

(a) Write down the equations of the asymptotes in this interval. 2 marks No calculator

Mark scheme

B1 x = 90°
B1 x = 270°

(b) Sketch the graph, showing the asymptotes correctly. 2 marks

Mark scheme

B1 correct tan shape with two branches, passing through (0,0), (180°,0) and (360°,0)
B1 asymptotes drawn as dashed vertical lines at 90° and 270°, curve not touching them

Question 3. Four curves are described. Curve P levels off towards $y = - 2$ at both ends. Curve Q shoots off vertically near $x = 4$ . Curve R is a straight line. Curve S levels off towards the $x$ -axis on the left and rises steeply on the right.

(a) Write down the equation of an asymptote for each of curves P, Q and S. 3 marks No calculator

Mark scheme

B1 P: y = -2
B1 Q: x = 4
B1 S: y = 0

(b) State why curve R has no asymptote. 1 mark No calculator

Mark scheme

B1 a straight line does not approach another line without meeting it oe

Words to know

Asymptote: A line that a curve approaches ever more closely but never touches or crosses.
Vertical asymptote: A line $x = a$ where the function is undefined, usually because a denominator is zero.
Horizontal asymptote: A line $y = b$ that the curve levels off towards as $x$ becomes very large or very negative.
Undefined: Having no value, as when a denominator is zero.

All Maths lessons Next: 3.6 Transforming graphs of functions