Calculus Small Changes, Big Ideas

Section 22.3 Limits of functions

• Let \(f:A\to\bb R\) and let \(c\) be a limit point of \(A\text{.}\) Then we say that

  \begin{equation*} \limit{x\to c}f(x)=L \end{equation*}

  if for all \(\eps>0\) there exists \(\delta >0\) such that for every \(x\in A\) for which \(0<|x-c|<\delta\text{,}\) we have

  \begin{equation*} |f(x)-L|<\eps\text{.} \end{equation*}

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• Sequential limits: \(\limit{x\to c}f(x)=L\) if and only if, for every sequence \(a_{n}\) from \(A\) for which each \(a_{n}\neq c\) and \(a_{n}\to c\text{,}\) we have \(f(a_{n})\to L\text{.}\)

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• Relate to continuity using \(\eps\)-\(\delta\) definition

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