# Edinburgh MATH08081 Introduction to Mathematical Analysis: Guide

> A study guide for MATH08081 at the University of Edinburgh: real numbers, sequences, series, limits, continuity and differentiation, with free practice.

Source: https://practicecalculus.com/courses/university-of-edinburgh-math08081
Updated: 2026-11-08

## At a glance

- **Course code:** MATH08081
- **Credits:** 20 SCQF credits (10 ECTS), SCQF Level 8
- **Taught:** Year 1, Semester 2
- **Prerequisites:** Introduction to Mathematics at University (MATH08078)
- **Assessment:** Written exam 80% (180 minutes), coursework 20%
- **Covers:** Real numbers, sequences, infinite series, limits, continuous functions, differentiation
- **Next course:** Further Analysis and Several Variable Calculus (MATH08082) in Year 2
- **Practice here:** [Calculus 1](/learn/calculus-1) plus [sequences and series](/learn/calculus-2/sequences-and-series/sequences)

## Key takeaways

- MATH08081 is a 20-credit, Year 1 course in rigorous analysis, the theory underneath calculus, taught in Semester 2.
- It covers real numbers, sequences, infinite series, limits, continuous functions and differentiation.
- Limits are written in epsilon-N and epsilon-delta language, and exponential, trig and log functions are defined rigorously.
- It is assessed by an 80% written exam and 20% coursework, and leads to MATH08082 in Year 2.

**MATH08081 Introduction to Mathematical Analysis** is the Year 1 analysis course in the School of Mathematics at the University of Edinburgh, taught in Semester 2. It is worth 20 SCQF credits (10 ECTS). The course catalogue calls analysis "the theory underpinning calculus": you prove why limits, continuity and derivatives behave the way you learned at school. This guide maps each MATH08081 topic to free practice so you can keep your calculus skills sharp while the proofs take centre stage.

## What MATH08081 covers

The Edinburgh course catalogue (DRPS) gives a representative outline: **real numbers, sequences, infinite series, limits, continuous functions and differentiation**. It says there is a strong focus on rigorous mathematics, "quite different to how mathematics is typically taught at school", building on the skills from Introduction to Mathematics at University (MATH08078) in Semester 1. It also says students will get plenty of chances to practise and reinforce their calculus skills.

The learning outcomes add detail. By the end you should understand the real numbers and the completeness axiom, limits in epsilon-N language, infinite sequences and series and their convergence, limits and continuity in epsilon-delta language, and differentiation. You also work with rigorous definitions of the exponential, trigonometric and logarithmic functions, and derive existence and uniqueness results for very simple linear ODEs with constant coefficients.

Compared with a North American **Calculus 1**, MATH08081 covers limits, continuity and derivatives plus sequences and series, but not integration, which comes in MATH08082. The emphasis is on proof.

## MATH08081 topic map

| MATH08081 topic | Practice | Read |
| --- | --- | --- |
| Real numbers and inequalities | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Absolute value function](/functions/absolute-value-function) |
| Sequences and their limits | [Sequences](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Infinite series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Convergence tests](/blog/convergence-tests), [geometric series](/blog/geometric-series) |
| Convergence tests | [Ratio and root tests](/learn/calculus-2/sequences-and-series/ratio-and-root-tests) | [Ratio test](/rules/ratio-test), [nth-term test](/rules/nth-term-test) |
| Limits of functions | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [Limit laws](/rules/limit-laws), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuous functions | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem), [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Differentiation from first principles | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Rules of differentiation | [Chain rule](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [product rule](/rules/product-rule) |
| Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Exponential, trig and log functions | [Exponential and log derivatives](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Natural exponential function](/functions/natural-exponential-function) |
| Simple linear ODEs | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Differential equations practice](/practice-problems/differential-equations) |

## Where MATH08081 fits

| Course | When | Focus |
| --- | --- | --- |
| MATH08078 Introduction to Mathematics at University | Year 1, Semester 1 | Proof and university-style mathematics (prerequisite) |
| MATH08081 Introduction to Mathematical Analysis | Year 1, Semester 2 | Sequences, series, limits, continuity, differentiation |
| MATH08082 Further Analysis and Several Variable Calculus | Year 2, Semester 2 | Rigorous integration, then multivariable calculus |

Engineering students take [Engineering Mathematics 1b (MATH08075)](/courses/university-of-edinburgh-math08075) instead, which is more computational and cannot be combined with MATH08081.

## Example problems at the MATH08081 level

**1. A limit of a sequence.** Find $\lim_{n \to \infty} \frac{3n^2 + 1}{n^2 + 5}$. Divide top and bottom by $n^2$:

$$\frac{3 + 1/n^2}{1 + 5/n^2} \to \frac{3}{1} = 3$$

In the course you would also prove it: for any $\varepsilon > 0$, find $N$ so that the distance to 3 is below $\varepsilon$ for all $n > N$.

**2. A convergence test.** Does $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converge? By the ratio test,

$$\frac{a_{n+1}}{a_n} = \frac{n+1}{2n} \to \frac{1}{2} < 1,$$

so the series converges.

**3. A derivative from the definition.** For $f(x) = x^2$,

$$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} (2x + h) = 2x$$

## How to do well in MATH08081

- **Keep computing while you learn to prove.** Proofs are much easier when the calculation behind them is automatic. Short daily sessions on [limits](/learn/calculus-1/limits/what-is-a-limit) and [series](/learn/calculus-2/sequences-and-series/choosing-a-test) help.
- **Learn the definitions word for word.** Most exam questions start from the precise definition of a limit, continuity or the derivative.
- **Work every tutorial sheet.** The catalogue stresses engaging with all the learning opportunities, not just lectures, and coursework is 20% of the mark.
- **Watch for common slips.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers the algebra errors that cost marks.

> **Tip:** the [squeeze theorem](/theorems/squeeze-theorem) is one of the most useful tools in a first analysis course. Practise spotting when a sequence can be trapped between two simpler ones.

## Frequently asked questions

### What is MATH08081 at Edinburgh?

MATH08081 Introduction to Mathematical Analysis is a first-year course from the School of Mathematics. The course catalogue describes it as a first course in rigorous mathematical analysis, covering real numbers, sequences, infinite series, limits, continuous functions and differentiation, while also practising calculus skills.

### What do I need before taking MATH08081?

The catalogue lists Introduction to Mathematics at University (MATH08078) as a prerequisite, which students take in Semester 1. Places are limited to students whose degree programme table includes the course.

### Can I take MATH08081 and Engineering Mathematics 1b together?

No. The catalogue lists Engineering Mathematics 1b (MATH08075) and Mathematics for the Natural Sciences 1b (MATH08073) as prohibited combinations with MATH08081.

### How is MATH08081 assessed?

Through a 180-minute written exam in the April/May diet worth 80% and coursework worth 20%. The catalogue says students must pass both the exam and the course overall.

### How can I prepare for MATH08081?

Make sure your limits, derivative rules and series tests are fluent, so the new effort goes into proofs rather than computation. The free Calculus 1 path and the sequences and series unit on PracticeCalculus cover the computational side.