Limit

Assume \(f(x)\) is a real function defined on some interval containing a point \(x_0\). We want to have the equivalent of a “tail property” for sequences – but this time for functions near \(x_0\). A property will be said to be a “local property” if whenever there exists a \(\delta>0\) such that \(f(x)=g(x)\) when \(|x-x_0|0\), \(f(x)=g(x)\) for every \(x\) such that 0<|x-x_0|0 such that for all \(x\) such that \(0<|x-x_0|0\) such that for the appropriate \(x\), \(f(x)<A\)” is an lpf property. Similarly, \(B<f(x)\) is an lpf property, and so is \(B<f(x)<A\). In that case, we will say that \(f\) is locally between \(B\) and \(A\).

Let \(C\) be a number, and assume that for every \(\epsilon\), \(f\) is between \(C-\epsilon\) and \(C+\epsilon\). As a synonym, we will also say that \(f\) has \(C\) as a limit at \(x_0\). In that case, we say that \(f\) tends to \(C\) as \(x\) tends to x_0. Making the definition more explicit: if for every \(\epsilon\) there exists a \(\delta\) such that if \(0<|x-x_0|<\delta\) then \(C-\epsilon<f(x)<C+\epsilon\) (or, equivalently, \(|f(x)-C|0\). We know that there is a \(\delta>0\), such that if \(0<|x-x_0|<\delta\), \(|f(x)-C|N\), \(0<|x_n-x_0|N\), \(|f(x_n)-C|0\), we see that \(f(x_n)\) is a sequence converging to \(C\). So we have: if \(f\) has a limit at x_0, \(f(x_n)\) will converge to the same limit if for every sequence \(x_n\) converging to \(x_0\).

Now, assume that for every sequence converging to \(x_0\), \(x_n\), \(f(x_n)\) converges to \(C\). Does that mean that \(f\) has a \(C\) as a limit at \(x_0\)? Assume the opposite. Then there is some \(\epsilon_0\) such that for every \(\delta\), we have some \(x\) such that \(0<|x-x_0|\epsilon_0\). For \(\delta=1/n\), take \(x_n\) to be one such \(x\). Then \(0<|x_n-x_0|\epsilon_0\), so that \(f(x_n)\) does not converge to \(C\), in contradiction to the original assumption. Therefore, if applying \(f\) to every sequence converging to \(x_0\) yields a sequence converging to \(C\), then \(f\) has a \(C\) as a limit at \(x_0\).

Because of that, we can use much of what we know of sequence convergence for function limits. For example, this implies that \(f+g\) will have as its limit the sum of the limits, and similarly for \(fg\) and \(f/g\).