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Understanding epsilon-delta proofs [Bautista] - mklabgr - 07-05-2026
Understanding epsilon-delta proofs
BY Guillermo Bautista
Summary
The article is a beginner-friendly explanation of ε–δ (epsilon–delta) proofs in calculus, aimed at first-year students who find limits confusing at first.
It starts by building intuition: expressions like ($|x-c| < \delta$) simply mean “x is close to c,” and ε–δ proofs are really about controlling how input closeness (x near c) forces output closeness (f(x) near L).
The key idea is the formal definition of a limit:
if you can show that for every tolerance ε (how close you want $f(x)$ to be to $L$), you can find a corresponding δ (how close x must be to c) so that the condition always holds, then the limit exists.
The article then walks through a simple example (a linear function). The main trick is:
you start from ($|f(x)-L| < ε$), rewrite it in terms of ($|x-c|$), and then “choose” δ as a function of ε (like $δ = ε/3$). This is not guessing randomly—it’s the core strategy: make the output condition depend on the input condition.
In the end, the article reassures the reader that $ε–δ$ proofs look scary but are actually systematic: manipulate algebra, connect ε and δ, and show the implication works for all $ε > 0$.
ARTICLE