MYTUTOR SUBJECT ANSWERS

429 views

How do I find the asymptotes of a curve?

The asymptotes of a curve are the lines or curves that approach a given curve very closely without actually touching it. They make curve sketching much easier. You will probably only have asymptotes that are lines in which case you will have either horizontal asymptotes or vertical asymptotes.

Horizontal asymptotes

To find the horizontal asymptotes, we want to see the behaviour of the curve as x tends to infinity. So we look at the order of both the numerator and denominator:

Case 1: If the numerator's order is greater than the denominator's (e.g if the function is f(x) = x3/(2x2+1) since xhas order 3 and x2 has order 2) then there will be no horizontal asymptotes.

Case 2: If the numerator's order is less than the denominator's (e.g if the function is f(x) = 1/x) then the horizontal asymptote will be y=0

Case 3: If the numerator's order is the same as the denominator's, e.g the function f(x) = (2x- 4) / (x2 - 1), then we look at the highest order term for both sides and look at the coefficient of it. In other words, we ignore all other terms besides 2x2 and x2 so 2x2/x2 = 2. So in this case the asymptote is y=2.

But why does this work? For the horizontal asymptotes we are effectively looking at what our curve will do as x tends to infinity. So in Case 1, when the numerator is bigger than the denominator, as x tends to infinity, y tends to infinity so there is no asymptote. In Case 2, as x tends to infinitiy, we find that y tends to zero. In case 3, as x tends to infinity, all of the smaller terms become insignificant. Let's take a closer look. In order to fully understand, we can substitute x=1000 (or any large number) to roughly see what happens.

Case 1: f(x) = x3/(2x2+1). Setting x=1000, we get                           f(1000)= 10003/(2(1000)2+1) = 500 and if we set x= 10000 then f(x)=5000 so there is no horizontal asymptote here.

Case 2: f(x) = 1/x. Setting x = 1000 we get f(1000)= 0.001 which is close to zero. Similarly, f(10000) = 0.0001 so as we increase x, f(x) tends to zero. Which is why the asymtote is y=0

Case 3:  f(x) = (2x- 4) / (x2 -1). f(1000) = (2(1000)- 4) / ((1000) -1)= 1.999998 which is roughly 2. If we increase x even more then f(10000)= 1.99999998 which is even closer to 2. Which is why the asymptote is y=2

Vertical asymptotes:

Here we are effectively looking at the behaviour of the curve as f(x)=y tends to infinity. The only times there will be vertical asymptotes is when the order of the denominator of the fraction is greater than the numerator's. Since f(x) will tend to infinity, the denominator of our curve will tend to zero. For example, let's consider the curve f(x) = (2x- 4) / (x2 -1)

Here we simply set the denominator x -1=0 to find x=1 or x= -1 and these are the vertical asymptotes.

Sobia K. A Level Further Mathematics  tutor, A Level Maths tutor, GCS...

1 year ago

Answered by Sobia, an A Level Further Mathematics tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist

68 SUBJECT SPECIALISTS

£20 /hr

Eden H.

Degree: Engineering Science (Masters) - Oxford, Somerville College University

Subjects offered: Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“Hi, I'm Eden, a fourth year engineering student at the University of Oxford. When I was taking my GCSEs my Maths teacher showed me how maths is the language we use to describe our reality, and ever since then I've been impassioned to ...”

£24 /hr

Oliver R.

Degree: Economics and Mathematics (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics
Maths

“About Me:I'm currently an undergraduate at the University of Bristol studying Economics & Mathematics (Joint Honours). I am one of those few who havea genuine love of studying Maths and hope that enthusiasm will help you in tutorials...”

£22 /hr

Chloe W.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics
Maths

“About Me:I'm Chloe and I'm a 2nd year maths student at Bristol university. I've always enjoyed working with numbers and I hope that I can encourage others to love working with them too!I've tutored both of my siblings through their...”

About the author

£24 /hr

Sobia K.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Further Mathematics , Spanish+ 1 more

Further Mathematics
Spanish
Maths

“Second year Maths Undergraduate, here to boost your grades!”

You may also like...

Other A Level Further Mathematics questions

How do I sketch the locus of |z - 5-3i | = 3 on an Argand Diagram?

How do you plot a complex number in an Argand diagram?

Find the set of values for which: 3/(x+3) >(x-4)/x

How do I find the asymptotes of a curve?

View A Level Further Mathematics tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok