## Limits

Calculus is one of the central branches of mathematics and was developed from algebra and geometry. It is built on the concept of limits, which will be discussed in this tutorial. Calculus consists of two related ideas: differential calculus and integral calculus. We will only be dealing with differential calculus in this tutorial and will explore how it can be used to solve optimisation problems and finding rates of change.

## The Tale of Achilles and the Tortoise

Zeno (born about 490 BC) was a philosopher of southern Italy who is famous for his paradoxes (a “paradox” is a statement that seems contradictory and yet may be true).

One of Zeno’s paradoxes can be summarised as:

Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles is very fast. So, the tortoise asks Achilles for a head start. Achilles agrees to give the tortoise a \(\text{1 000}\) \(\text{m}\) head start. Does Achilles overtake the tortoise?

To solve this problem, we start by writing:

\begin{align*} \text{Achilles:} \quad {x}_{A} & = {v}_{A}t \\ \text{Tortoise:} \quad {x}_{T} & = \text{1 000}\text{ m} + {v}_{T}t \end{align*}

where

- \({x}_{A}\) is the distance covered by Achilles
- \({v}_{A}\) is Achilles’ speed
- \(t\) is the time taken by Achilles to overtake the tortoise
- \({x}_{T}\) is the distance covered by the tortoise
- \({v}_{T}\) is the tortoise’s speed

Achilles will overtake the tortoise when both of them have covered the same distance. If we assume that Achilles runs at \(\text{2}\text{ m.s$^{-1}$}\) and the tortoise runs at \(\text{0.25}\text{ m.s$^{-1}$}\), then this means that Achilles will overtake the tortoise at a time calculated as:

\begin{align*} {x}_{A} & = {x}_{T} \\ {v}_{A}t & = \text{1 000} + {v}_{T}t \\ 2 t &= \text{1 000} + \text{0.25}t \\ 2 – \text{0.25}t &= \text{1 000} \\ \cfrac{7}{4}t &= \text{1 000} \\ t &= \cfrac{\text{4 000}}{7} \\ &= \text{571.43}\text{ s} \end{align*}

However, Zeno looked at it as follows:Achilles takes \(t=\cfrac{\text{1 000}\text{ m}}{\text{2}\text{ m.s$^{-1}$}}= \text{500}\text{ s}\) to travel the \(\text{1 000}\text{ m}\) head start that he gave the tortoise. However, in these \(\text{500}\text{ s}\), the tortoise has travelled a further \(x = \text{500}\text{ s} \times \text{0.25}\text{ m.s$^{-1}$} = \text{125}\text{ m}\).

Achilles then takes another \(t=\cfrac{\text{125}\text{ m}}{\text{2}\text{ m.s$^{-1}$}}= \text{62.5}\text{ s}\) to travel the \(\text{125}\text{ m}\). In these \(\text{62.5}\) \(\text{s}\), the tortoise travels a further \(x = \text{62.5}\text{ s} \times \text{0.25}\text{ m.s$^{-1}$} = \text{15.625}\text{ m}\).

Zeno saw that Achilles would always get closer and closer but wouldn’t actually overtake the tortoise.

So what does Zeno, Achilles and the tortoise have to do with calculus? Consider our earlier studies of sequences and series:

We know that the sequence \(0;\cfrac{1}{2};\cfrac{2}{3};\cfrac{3}{4};\cfrac{4}{5}; \ldots\) can be defined by the expression \({T}_{n}=1-\cfrac{1}{n}\) and that the terms get closer to \(\text{1}\) as \(n\) gets larger.

Similarly, the sequence \(1;\cfrac{1}{2};\cfrac{1}{3};\cfrac{1}{4};\cfrac{1}{5};\ldots\) can be defined by the expression \({T}_{n}=\cfrac{1}{n}\) and the terms get closer to \(\text{0}\) as \(n\) gets larger.

We have also seen that an infinite geometric series can have a finite sum.

\[{S}_{\infty }=\sum _{i=1}^{\infty }{a}.{r}^{i-1}=\cfrac{{a}}{1-r} \quad \text{ for }-1<r<1\]

where \(a\) is the first term of the series and \(r\) is the common ratio.

We see that there are some functions where the value of the function gets close to or **approaches** a certain value as the number of terms increases.