Calculus Small Changes, Big Ideas

Section 5.1 Limits and continuity

• Introduce limits as a way to ‘clean up’ the infinitesimal arguments we’ve been making so far.

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• Examine the cardinal sine function

  \begin{equation*} \sinc x=\frac{\sin x}{x} \end{equation*}

  around \(x=0\text{.}\) As \(x\to 0\text{,}\) \(\sinc x\to 1\text{,}\) so we say

  \begin{equation*} \limit{x\to 0}\frac{\sin x}{x}=1 \end{equation*}

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• Avoid the epsilon-delta definition at this stage... this can go in Part V. But we can perhaps still talk about ‘tolerance’ or ‘arbitrary closeness.’ Might even be able to use the symbols \(\delta\) and \(\eps\) for tolerance in the \(x\)- and \(y\)-directions, but hold back from the full formal definition.

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• Use the the Heaviside step function

  \begin{equation*} H(x)=\begin{cases}0 \amp \text{if }x\lt 0\\ 1 \amp \text{if }x\ge 0\end{cases} \end{equation*}

  to discuss one-sided limits

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• Continuity in terms of limits: \(f\) is continuous at \(x=a\) if

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

  This means the function must be defined, the limit must exist, and the function must match the limit.

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• Types of discontinuity: removable, jump, infinite, oscillating

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• Importantly, do NOT throw away the infinitesimal ideas just because we’re using limits now! We’ll see the relationship between infinitesimals and everything we’ve done so far in more detail in Part V.

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