UCL Online STEP and AEA Preparation Sessions
Session 13: Analysing the behaviour of functions – limits and curve
sketching
Curve sketching is a very important skill in all mathematical subjects – from
Economics to Engineering. Algebraic techniques such as factorising, long division
and the binomial expansion can be very useful to determine the behaviour of
functions. Of particular interest in this session will be limiting values of functions and
their behaviour around asymptotes. For example it’s clear that the function
f ( x) 
x2 1
x2 1
is not defined at x = 1, but is there such a thing as lim
?
x 1 x  1
x 1
This session will cover techniques for analysing questions like this as well as for
general curve sketching.
Having some knowledge of how quickly functions approach infinite (polynomial vs
exponential growth for example) is useful.
The session will also cover understanding graphs through the use of elementary
transformations and changes of variable.
Example problem
What is lim
x  27
x  27
1
3
x 3
? Explain why this is the case.