1.2 Two Useful Results

One thing we would definitely like to know is that if a limit of a function exists, is it unique? Luckily, the answer to this is yes, and is summed up in the following result:

Proposition 1.2: (Uniqueness of Limits)

Let \(c\in\mathbb{R}\) and let \(f:D \to \mathbb{R}\) be a function defined on a punctured neighbourhood \(D\) of \(c\). If \(\exists L, M \in \mathbb{R}\) such that \(\lim_{x\to c} f(x) = L\) and \(\lim_{x\to c} f(x) = M\), then \(L=M\).

Another useful result regarding functions we can state here is known as the inertia principle:

Proposition 1.3: (Inertia)

Let \(c\in\mathbb{R}\) and let \(f:D \to \mathbb{R}\) be a function defined on a punctured neighbourhood \(D\) of \(c\). If \(\lim_{x\to c} f(x) = L\), then for any \(M < L\), there exists \(\delta > 0\) such that \[\forall x \in D,\;\; 0 < \lvert x - c \rvert < \delta \; \Rightarrow \;\; f(x) > M.\]

This idea is summed up in Figure 1.2 below. This principle is especially useful when the limit of a function at a point \(c\) is positive, as it says that there is a punctured neighbourhood around \(c\) in which the function remains positive.

Figure 1.2: A diagram showing the inertia principle in action. This principle says that if a function approaches a limit at a point, there is always an interval in which the function stays within a specified distance of the limit