MATHEMATICAL ANALYSIS 03
Limits More Rigorously
The epsilon-delta definition makes “approaches” mathematically precise.
The definition
\[\boxed{\lim_{x\to a}f(x)=L}\]
means that for every \(\varepsilon\gt0\), there exists \(\delta\gt0\) such that
\[0\lt|x-a|\lt\delta\quad\Longrightarrow\quad |f(x)-L|\lt\varepsilon.\]What the symbols mean
\(\varepsilon\) specifies how close the output must be to \(L\). We choose \(\delta\) so that inputs sufficiently close to \(a\) force that output accuracy.
Example
For \(f(x)=2x\), to prove \(\lim_{x\to3}2x=6\), note
\[|2x-6|=2|x-3|.\]Choosing \(\delta=\varepsilon/2\) makes \(|x-3|\lt\delta\) imply \(|2x-6|\lt\varepsilon\).
Why this matters
Derivatives, continuity and integrals ultimately depend on limits. The definition tells us exactly what those limiting statements mean.