The Precise Definition of a Limit

The informal limit definition says $f(x)$ gets "arbitrarily close to $L$" when $x$ is "sufficiently close to $a$". To prove limit laws, to distinguish a genuine limit from a numerical artifact, and to verify convergence of a numerical method (such as a binomial tree), you need quantitative control: how close must $x$ be to guarantee that $f(x)$ is within a prescribed tolerance? The $\varepsilon$-$\delta$ definition answers that question. It turns the limit concept into something provable and falsifiable, and its structure — challenge–response, universal–existential — is the template for every convergence proof in analysis.

• state the $\varepsilon$-$\delta$ definition precisely and interpret both quantifiers;
• use the $\delta$-chase strategy to verify that a given limit claim is correct;
• write a complete $\varepsilon$-$\delta$ proof for a linear function and for a simple quadratic;
• translate the definition into a geometric picture (an $\varepsilon$-strip around $L$ forces an $x$-interval around $a$).

1. The $\varepsilon$-$\delta$ definition
2. The $\delta$-chase strategy
3. Infinite limits and vertical asymptotes