1.3 Sequential Characterisation of Limits
Using an \(\epsilon\)-\(\delta\) definition of the limit is not the only way we can define the limit of a function! In fact, you’ve already seen something similar when dealing with sequential continuity last semester.
Let \(c\in\mathbb{R}\) and let \(f: D \to \mathbb{R}\), with \(D\) a punctured neighbourhood of \(c\). Then the following are equivalent2:
\(\lim_{x\to c}f(x) = L,\)
For any sequence \((x_n)_n\) in \(D\setminus\lbrace c \rbrace\) which converges to \(c\), \[\lim_{n\to\infty}f(x_n) =L.\]
Why should you be pleased about this theorem? This allows you to apply all your Semester 1 knowledge of sequences to functions (for example, the algebra of limits)! In particular, if you can find two sequences \((x_n)_n\) and \((y_n)_n\) in \(D\setminus\lbrace c\rbrace\) converging to \(c\), but the sequences \(\left(f(x_n)\right)_n\) and \(\left(f(y_n)\right)_n\) approach different limits, this gives you an easy way of proving that function limits do not exist.
You may see this abbreviated to ‘TFAE’ in some contexts.↩︎