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Calculus III

Course: Calculus I
Topic: Limits and Continuity
Subtopic: Properties of Limits

Overview

This lesson makes formal the properties of limits (a.k.a. limit theorems). These properties govern what you can and cannot do algebraically when evaluating a limit. For instance you can rewrite a limit of a sum as a sum of limits, but you cannot rewrite a limit of a product as a product of limits.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 1.5.1 Understand the properties of limits
• 1.5.2 Evaluate limits algebraically by correctly applying the appropriate properties of limits
• 1.5.3 Understand, know when to apply, and know how to properly use the squeeze theorem
• 1.5.4 Evaluate limits of transcendental functions including trigonometric, exponential, and logarithmic
• 1.5.5 Recognize "special limits" such as the limit of (sinx)/x as x->0

Terminology

Terms you should be able to define: squeeze theorem (a.k.a. sandwich theorem)

Text Notes

• The Squeeze Theorem, also known as The Sandwich Theorem, is so named because you are squeezing the limit between two curves like a sandwich. It is sometimes also called the Pinching Theorem so clearly there are a variety of names all based on the same squishing concept.
• About four "special limits" have been introduced (or will be soon) that are worth memorizing (below). We'll prove/justify them sometime in this chapter.

Special Limits

 lim_(x->0) sinx/x = 1 lim_(x->0) tanx/x = 1 lim_(x->0) (1-cosx)/x = 0 lim_(x->0) (1+x)^(1/x) = e

Mini-Lectures and Examples

STUDY: Limits - Algebraically

Supplemental Resources (optional)

Geogebra Exploration: The Limit Laws, Mark Renault

Geogebra Exploration: A Special Limit, Mark Renault

rev. 2020-10-10