Limits — ε-δ Definition & Computation

Intuitive "closeness" works for textbook examples but breaks on sneakier functions like $\sin(1/x)$, the Dirichlet function, or limits of sequences of functions. The ε-δ definition gives a single, machine-checkable criterion that resolves every ambiguity. It also turns "the limit exists" from a vague feeling into a logical statement with a quantifier order ($\forall\eps\,\exists\delta$). Once you have this, every later theorem (continuity, differentiability, integrability, uniform convergence) becomes provable instead of hand-wavy.

The horizontal orange band is the ε-tube: $\{y : \abs{y-L}<\eps\}$. The vertical green strip is the δ-window: $\{x : 0<\abs{x-a}<\delta\}$. The ε-δ condition holds iff the part of the curve $y=f(x)$ above the green strip lies entirely inside the orange band. If even one point of the curve pokes above or below the band while $x$ is in the strip, the condition fails — the readout flips to FAIL. Toggling Auto-δ runs a bisection that returns the largest δ for which the condition holds.

Everywhere we need guaranteed precision. In numerical computing, ε is your tolerance and δ is the input precision needed to achieve it — this is the language of error analysis (e.g. when you specify atol=1e-9 in an ODE solver). Control engineers use it for stability margins (a controller is "robust" iff a δ-perturbation in the plant gives only an ε-perturbation in output). In machine learning, generalization bounds are ε-δ statements about test error. Even in physics, "the wave function is continuous" means an ε-δ statement about the field.

Because the limit must not depend on what happens at $a$ — only what happens around it. The function may not even be defined at $a$ (think $f(x)=\sin(x)/x$ at $x=0$), or it may be defined with a "wrong" value (a removable discontinuity). The punctured condition $0<\abs{x-a}$ excludes $x=a$ from the test, so $f(a)$ is irrelevant to the limit. This is also why a limit can exist even where $f$ is undefined, and why continuity at $a$ ($\lim_{x\to a}f(x)=f(a)$) is a separate, stronger property.

For polynomial $f$ of degree $n$ near $a$, Taylor expansion gives $\abs{f(x)-L} \approx \abs{f'(a)}\,\abs{x-a} + O(\abs{x-a}^2)$, so $\delta \approx \eps / \abs{f'(a)}$ works to first order. For Lipschitz $f$ with constant $K$, $\delta = \eps/K$ always works. For $f$ with a removable discontinuity (like $\sin(x)/x$), use a Taylor estimate: near $0$, $\sin(x)/x \approx 1 - x^2/6$, so $\delta \approx \sqrt{6\eps}$ — note δ is $O(\sqrt{\eps})$, not $O(\eps)$. The complexity of the numerical bisection in the simulation is $O(\log_2(\delta_{\max}/\delta_{\min}))$ — about 25 steps to reach $\eps\approx 10^{-7}$.

Re-read the definition: "$\abs{x-a}<\delta$" is "$x$ is in an open ball of radius $\delta$ around $a$"; "$\abs{f(x)-L}<\eps$" is "$f(x)$ is in an open ball of radius $\eps$ around $L$". The ε-δ statement is exactly: for every open ball around $L$, the preimage contains an open punctured ball around $a$. This generalizes verbatim to arbitrary topological spaces by replacing "open ball" with "open set" — yielding the topological definition: $f$ is continuous at $a$ iff the preimage of every open set containing $f(a)$ contains an open set containing $a$. So ε-δ is the metric-space version of an open-set statement.

Negate the definition. "$\lim_{x\to a}f(x)=L$ holds" $=$ "$\forall\eps\,\exists\delta\,\forall x:\,0<\abs{x-a}<\delta\Rightarrow\abs{f(x)-L}<\eps$." The negation is "$\exists\eps\,\forall\delta\,\exists x:\,0<\abs{x-a}<\delta\,\wedge\,\abs{f(x)-L}\geq\eps$." Read it: "there is a tolerance $\eps$ that I can never achieve, no matter how small a window I pick." For $\sgn(x)$ at $x=0$ with proposed $L=0$, take $\eps = 1/2$ — for any $\delta$, the point $x=\delta/2$ satisfies $\abs{f(x)-0}=1\geq 1/2$. So no $L$ works, and the limit does not exist.