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Section 3.2 The formal definition of limits

The informal definition of limits given in the previous subsection is imprecise, since we have not defined what it means to say “arbitrarily close” and “sufficiently close”. The formal definition of limits makes such statements precise.

Subsection 3.2.1 Instructional video

There is no instructional video on this particular topic. It will be covered during the live lecture.

Subsection 3.2.2 Key concepts

Concept 3.2.1. The formal definition of limits.

Let \(a \in \mathbb{R}\text{,}\) and let \(f\) be a function defined on some open interval that contains \(x=a\text{,}\) except possibly at \(x=a\) itself. Then we write

\begin{equation*} \lim_{x \to a} f(x) = L, \end{equation*}

and say that the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) is \(L\), if and only if for every real number \(\epsilon > 0\text{,}\) there exists a real number \(\delta > 0\) such that:

\begin{equation*} \text{if } 0 \lt |x-a| \lt \delta, \text{ then } |f(x) - L | \lt \epsilon. \end{equation*}

Since \(\epsilon\) is an arbitrary positive number, what this says is that we can make the distance between \(f(x)\) and \(L\) arbitrarily small, by taking the distance between \(x\) and \(a\) sufficiently small.

Further readings 3.2.3 Further readings