Calculus I · Limits and Continuity · review
Epsilon-Delta Review
Section 6 Summary
• is an allowed output error; is an input tolerance that guarantees it. • Formal proofs work backward to discover and forward to verify it. • Linear proofs usually use . • Nonlinear proofs often need a preliminary local bound and a minimum of two restrictions. • To disprove a claimed limit, one bad is enough.
After the explanation
Use the section idea
How small must the input window be to force every allowed output into the requested tolerance band?
Epsilon sets the demanded vertical accuracy; delta is the horizontal promise you choose so every permitted nearby input meets that demand.
Work backward from the desired output inequality, isolate an input-distance bound, then state a positive delta that is no larger than that bound.
A proof must control every eligible input in the punctured window; checking examples or choosing delta after seeing the input is not enough.
Formal understanding means you can translate between bands, inequalities, and words, then verify the implication from delta to epsilon in forward order.
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