Assumed knowledge

The content of the modules:

Motivation

Functions are the heart of modelling real-world phenomena. They show explicitly the relationship between two (or more) quantities. Once we have such a relationship, various questions naturally arise.

For example, if we consider the function

$f(x) = \dfrac{\sin x}{x},$

we know that the value $$x=0$$ is not part of the function's domain. However, it is natural to ask: What happens near the value $$x=0$$? If we substitute small values for $$x$$ (in radians), then we find that the value of $$f(x)$$ is approximately 1. In the module The calculus of trigonometric functions, this is examined in some detail. The closer that $$x$$ gets to 0, the closer the value of the function $$f(x) = \dfrac{\sin x}{x}$$ gets to 1.

Another important question to ask when looking at functions is: What happens when the independent variable becomes very large? For example, the function $$f(t)= e^{-t}\sin t$$ is used to model damped simple harmonic motion. As $$t$$ becomes very large, $$f(t)$$ becomes very small. We say that $$f(t)$$ approaches zero as $$t$$ goes to infinity.

Both of these examples involve the concept of limits, which we will investigate in this module. The formal definition of a limit is generally not covered in secondary school mathematics. This definition is given in the Links forward section. At school level, the notion of limit is best dealt with informally, with a strong reliance on graphical as well as algebraic arguments.

When we first begin to teach students how to sketch the graph of a function, we usually begin by plotting points in the plane. We generally just take a small number of (generally integer) values to substitute and plot the resulting points. We then join the dots'. That we can join the dots' relies on a subtle yet crucial property possessed by many, but not all, functions; this property is called continuity. In this module, we briefly examine the idea of continuity.

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