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Calculus I / Limits and Continuity

Derivation — ε-δ Proofs of the Sum and Product Limit Laws

The limit laws in Calculating Limits were stated without proof. This derivation note supplies the ε\varepsilon-δ\delta proofs for the two load-bearing laws — the sum law and the product law — and sketches how the quotient and power laws follow. These proofs are not merely certification exercises: the δ\delta-budget idea (split ε\varepsilon among several terms) is the template for most multi-step limit arguments in analysis, including the proof that every composition of continuous functions is continuous.
Learning objectives. By the end of this note you should be able to:
  • write a complete ε\varepsilon-δ\delta proof of the sum limit law;
  • identify and apply the ε/2\varepsilon/2 splitting trick;
  • write a complete proof of the product limit law including the auxiliary bound on ff;
  • state without proof how the quotient and power laws are derived from the sum and product.

In this lesson

  1. The sum law
  2. The product law
  3. Extension to quotient and power

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