Rigorously Defining the Limit Using the Epsilon Delta Definition

Despite being the foundation for most concept in calculus, in most high school calculus courses, the limit is often presented as an ambiguous concept without a clear definition. However, there does actually exist a rigorous definition known as the ‘epsilon delta definition’ and is usually not taught until early university courses in calculus or analysis. Therefore, for this article, I would like to go through this definition and show how we can use it to prove certain limits.

The Epsilon-Delta Definition

The main objective of the epsilon-delta definition is to give a proper meaning to the following statement:

For some function f : A → R. One intuitive explanation some may use is that ‘as x gets infinitesimally closer to c, f(x) gets infinitesimally closer to L’, but this definition is quite hand-wavy and unclear. Instead, a more accurate definition using the epsilon-delta definition would be the following:

Let f: A → R. lim(x → c) = L means that “∀ϵ > 0, ∃δ > 0 such that 0 < |x - c| < δ (where x ∈ A) ⟹ |f(x) - L| < ϵ”

If you do not understand the symbols, ∀ means ‘for all, ∃ means ‘there exists’, and ⟹ mean ‘implies’. So, to put this into words, the epsilon delta definition means that “for all ϵ > 0, there exists a…