Exercise 1153Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

The $\varepsilon$-$\delta$ definition of limit

#epsilon-delta#quantifiers#limit

Statement

Write the definition of $\lim_{x\to a}f(x)=\ell$ using quantifiers, and write its negation.

Hint

The positive definition uses $\forall\varepsilon>0,\ \exists\delta>0,\ \forall x,\ \ldots$

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Definition.
$\forall\varepsilon>0,\ \exists\delta>0,\ \forall x\in\mathbb{R},\ (0<|x-a|<\delta\Rightarrow|f(x)-\ell|<\varepsilon).$

Negation.
$\exists\varepsilon>0,\ \forall\delta>0,\ \exists x\in\mathbb{R},\ (0<|x-a|<\delta\ \text{and}\ |f(x)-\ell|\ge\varepsilon).$

Conclusion. The negation expresses "there is some $\varepsilon$ such that, however small $\delta$ is, some $x$ nearby violates the bound".