Limits using Series (Edexcel A Level Further Maths): Revision Note

Exam code: 9FM0

Author

Mark Curtis

Last updated

6 January 2026

Limits using series

How do I find limits using Taylor series?

To find the limit of a function as $x$ tends to $a$ , $\lim_{x \to a} f (x)$
- first write $f (x)$ as a Taylor series about $x = a$
- then let $x \to a$ in each term
  - Each ${(x - a)}^{n}$ term becomes zero leaving behind the limit
This method works for combinations of functions
- e.g. $\lim_{x \to a} (\frac{f (x)}{g (x)})$ , $\lim_{x \to a} (f (x) \pm g (x))$ or $\lim_{x \to a} (f (x) g (x))$
  - First write $f (x)$ and $g (x)$ as a Taylor series about $x = a$
  - Simplify the algebra inside the limit
  - Then let $x \to a$ and see what is left behind

Examiner Tips and Tricks

You will be given the Taylor series expansion in the exam question:

$f (x) = f (a) + (x - a) f' (a) + \frac{{(x - a)}^{2}}{2!} f'' (a) + . . . + \frac{{(x - a)}^{r}}{r!} f^{(r)} (a) + . . .$

How do I find limits using Maclaurin series?

In the special case when $a = 0$
- the limit is $\lim_{x \to 0} f (x)$
- you can use Maclaurin series
  - $f (x) = f (0) + x f' (0) + \frac{x^{2}}{2!} f'' (0) + . . . + \frac{x^{r}}{r!} f^{(r)} (0) + . . .$
This is particularly useful for functions inside functions
- e.g. $\lim_{x \to 0} (\frac{\sin (x^{3})}{x^{3}})$
  - use $\sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - . . .$ from the formulae booklet
- substitute in $x^{3}$
  - $\sin (x^{3}) = x^{3} - \frac{x^{9}}{3!} + \frac{x^{15}}{5!} - . . .$
- cancel and take the limit
  - $\lim_{x \to 0} (\frac{x^{3} (1 - \frac{x^{6}}{3!} + \frac{x^{12}}{5!} -)}{x^{3}}) = 1 - 0 + 0 - . . . = 1$

Examiner Tips and Tricks

You are given the Maclaurin series formula in the formulae booklet, as well as Maclaurin series expansions for $e^{x}$ , $\ln (1 + x)$ , $\sin x$ , $\cos x$ and $\arctan x$ .

Worked Example

By finding the Taylor series expansion about $x = 1$ of $\ln x$ in ascending powers of $(x - 1)$ , up to and including the term in ${(x - 1)}^{4}$ , find

$\lim_{x \to 1} (\frac{\ln x - (x - 1) + \frac{1}{2} {(x - 1)}^{2}}{{(x - 1)}^{3}})$

Answer:

First find the Taylor series by letting $f (x) = \ln x$ and finding the first four derivatives

$\begin{array}{rcl} f' (x) & = & \frac{1}{x} \\ f'' (x) & = & - \frac{1}{x^{2}} \\ f''' (x) & = & \frac{2}{x^{3}} \\ f^{(4)} (x) & = & - \frac{6}{x^{4}} \end{array}$

Substitute $x = 1$ into $f (x)$ and its derivatives

$\begin{array}{rcl} f (1) & = & \ln 1 = 0 \\ f' (1) & = & \frac{1}{1} = 1 \\ f'' (1) & = & - \frac{1}{1^{2}} = - 1 \\ f''' (1) & = & \frac{2}{1^{3}} = 2 \\ f^{(4)} (1) & = & - \frac{6}{1^{4}} = - 6 \end{array}$

Substitute these values into the Taylor series expansion given by

$f (x) = f (a) + (x - a) f' (a) + \frac{{(x - a)}^{2}}{2!} f'' (a) + . . . + \frac{{(x - a)}^{r}}{r!} f^{(r)} (a) + . . .$

$\begin{array}{rcl} \ln x & = & 0 + (x - 1) \times 1 + \frac{{(x - 1)}^{2}}{2!} \times (- 1) + \frac{{(x - 1)}^{3}}{3!} \times 2 + \frac{{(x - 1)}^{4}}{4!} \times (- 6) + . . . \\ = & (x - 1) - \frac{1}{2} {(x - 1)}^{2} + \frac{1}{3} {(x - 1)}^{3} - \frac{1}{4} {(x - 1)}^{4} + . . . \end{array}$

Now substitute this Taylor series into the expression in the question, cancelling any terms

$\frac{\ln x - (x - 1) + \frac{1}{2} {(x - 1)}^{2}}{{(x - 1)}^{3}} = \frac{(x - 1) - \frac{1}{2} {(x - 1)}^{2} + \frac{1}{3} {(x - 1)}^{3} - \frac{1}{4} {(x - 1)}^{4} + . . . - (x - 1) + \frac{1}{2} {(x - 1)}^{2}}{{(x - 1)}^{3}}$

Factorise out ${(x - 1)}^{3}$ from top and bottom and cancel

$\begin{array}{rcl} = & \frac{\frac{1}{3} {(x - 1)}^{3} - \frac{1}{4} {(x - 1)}^{4} + . . .}{{(x - 1)}^{3}} \\ = & \frac{{(x - 1)}^{3} (\frac{1}{3} - \frac{1}{4} (x - 1) + . . .)}{{(x - 1)}^{3}} \\ = & \frac{1}{3} - \frac{1}{4} (x - 1) + . . . \end{array}$

Now take the limit as $x \to 1$ (i.e. all terms involving ${(x - 1)}^{n}$ go to zero)

$\begin{array}{rcl} \lim_{x \to 1} (\frac{\ln x - (x - 1) + \frac{1}{2} {(x - 1)}^{2}}{{(x - 1)}^{3}}) & = & \lim_{x \to 1} (\frac{1}{3} - \frac{1}{4} (x - 1) + . . .) \\ = & \frac{1}{3} + \lim_{x \to 1} (- \frac{1}{4} (x - 1) + . . . .) \\ = & \frac{1}{3} + 0 \\ = & \frac{1}{3} \end{array}$

Write out the final result

$\lim_{x \to 1} (\frac{\ln x - (x - 1) + \frac{1}{2} {(x - 1)}^{2}}{{(x - 1)}^{3}}) = \frac{1}{3}$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent

Limits using Series (Edexcel A Level Further Maths): Revision Note

Limits using series

How do I find limits using Taylor series?

How do I find limits using Maclaurin series?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Further Trigonometry

The t-formulae

Trig identities using t-formulae

Trig Equations using t-formulae

Modelling using t-formulae

Further Calculus

Taylor Series

Taylor Series

Limits using Series

Series Solutions to Differential Equations

Methods in Calculus

Leibnitz's Theorem

L'Hospital's Rule

The Weierstrass Substitution

Further Differential Equations

Reducing Differential Equations

Reducing First-Order Differential Equations

Reducing Second-Order Differential Equations

Coordinate Systems

Properties of Ellipses, Parabolas & Hyperbolas

Properties of Ellipses

Properties of Parabolas

Properties of Hyperbolas

Properties of Rectangular Hyperbolas

Tangent & Normals to Ellipses, Parabolas & Hyperbolas

Tangents & Normals to Ellipses

Tangents & Normals to Parabolas

Tangents & Normals to Hyperbolas

Tangents & Normals to Rectangular Hyperbolas

Loci Problems

Loci Problems

Further Vectors

Further Vectors

Vector (Cross) Product

Scalar Triple Product

Vector Geometry with Points, Lines & Planes

Shortest Distances using Vector Rules

Areas & Volumes using Vector Rules

Direction Ratios & Direction Cosines

Further Numerical Methods

Further Numerical Methods

Numerical Solutions of First-Order Differential Equations

Numerical Solutions of Second-Order Differential Equations

Simpson's Rule

Inequalities

Inequalities

Solving Inequalities involving Algebraic Fractions

Solving Inequalities involving Modulus Functions

Author: Mark Curtis