diff --git a/content/history/set-theory/limits.tex b/content/history/set-theory/limits.tex index 02dbf8cb..4944f02a 100644 --- a/content/history/set-theory/limits.tex +++ b/content/history/set-theory/limits.tex @@ -34,8 +34,16 @@ making $x$ suitably close to~$c$. More precisely, we stipulate that $\lim_{x \rightarrow c} g(x) = \ell$ will mean: \[ -(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(|x - c| < \delta \lif |g(x) - \ell| < \epsilon \right). +(\exists\ell \in \Real)(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(0 < |x - c| < \delta \lif |g(x) - \ell| < \epsilon \right). \] +Here $\ell$ is unique. That is, if both $\ell_1 \in \Real$ and +$\ell_2 \in \Real$ witness the above sentence, it must be the case +that $\ell_1=\ell_2$. We say that the limit of a function is unique. + +\begin{prob} + Prove that the limit of a function is unique. +\end{prob} + The vertical bars here indicate absolute magnitude. That is, $|x| = x$ when $x \geq 0$, and $|x| =-x$ when $x < 0$; you can depict \emph{that} function as follows: