Chapter 4 Limits of Functions
"But wait!" I hear you say, "We were supposed to be doing analysis of functions! We don’t have any functions yet (except if we think about sequences as functions, which to be honest we really haven’t been.)"
You’re right! Let’s fix that. First on the agenda are limits.
We’re going to be working with functions \(f:\mathbb{R}\to\mathbb{R}\text{,}\) \(f:A\to \mathbb{R}\) for \(A\subseteq \mathbb{R}\text{,}\) and most of what we’ll say will carry over into functions \(F:K\to W\) where \(K\subseteq V\) is a subset of a normed space \((V,\lVert\cdot\rVert_V)\) and \((W,\lVert\cdot\rVert_W)\) is a normed space.
The basic idea of a function limit is this: we write
\begin{equation*}
\displaystyle\lim_{x\to c}f(x)=L
\end{equation*}
if we can force \(f(x)\) to be arbitrarily close to \(L\) simpy by guaranteeing \(x\) is sufficiently close to \(c\text{.}\)
