Limits part 2: Functions: All kinds of limits
Limits at a point
Instead of studying the behaviour of a function for \(x\to\infty\), we can analyse functions, for example, around a chosen fixed point, say \(x=a\). This means that we consider the values of \(f(x)\) for \(x\) values near \(a\).
The limit of a function \(f(x)\) as \(x\) approaches \(a\) is \(L\) if for every number \(\epsilon > 0\) there exists a number \(\delta > 0\) such that if \(0<|x - a|<\delta\) then \(x\) belongs to the domain of \(f\) and \(|f(x) - L|<\epsilon\).
This is denoted by \[ \lim_{x\to a} f(x) = L\text. \] The \(\epsilon\) in the definition can again be regarded as a margin of error. Given the margin of error there exists a small \(\delta\) such that that for all \(x\in \left( a - \delta, a + \delta \right)\) the difference between \(f(x)\) and limit \(L\) is smaller than the margin of error. This means that for \(x\) values nearby \(a\) the function values are very close to \(L\). Note that \(f(a)\) dies not need to exist. Even if \(f(a)\) exists, then it does not have to be equal to \(L\), because the point \(x=a\) does not satisfy \(0<|x-a|\).
It is especially interesting to study functions around a point when they are not defined at that point. Study some examples by clicking on "new example"
After inspection of a couple of examples, you have seen limits being computed for rational function that were not defined in the point of interest because both the numerator and denominator were zero in this point. The limits are therefore of the indeterminate form \(\frac{0}{0}\) when the limit point is substituted in the function description.
By factorisation of the numerator and denominator one can sometimes cancel terms after which one is able to apply the limit rules.
In the section Techniques we devote a paragraph on how one can decompose functions into factors.
