Fundamentals of University Mathematics, 3rd Edition

Appendix C

Limits and Continuity—A Rigorous Approach

In this appendix we introduce the formal definitions of function limit, continuity and sequence limit. Their application is illustrated with examples and a selection of theorems. A proof of L’Hôpital’s rule is also included.

C.l Function Limits

Definition C.1.1 Let f be a real function whose domain contains a punctured neighbourhood of c and let L ∈ . Then the limit of f at c exists and equals L, and we write

if and only if, for each ε > 0, there exists δ > 0 such that

Notes C.l.2

(1) In general, ...