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The Calculus Lifesaver by Adrian Banner

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APPENDIX A ______________________________

Limits and Proofs

Throughout this book we have used limits extensively, in their own right and also as an essential part of the definitions of the derivative and the integral. Since limits are so important, it’s about time that we define them properly. Once we know how they work, we can prove a number of facts that we’ve been taking for granted. So, here’s what’s in this appendix:

• the formal definition of a limit (including left-hand and right-hand limits, infinite limits, limits at ±∞, and limits of sequences);

• combining limits, and a proof of the sandwich principle;

• the relationship between continuity and limits, including a proof of the Intermediate Value Theorem;

• differentiation and limits, ...

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