Stage 4 Lesson Hub

This Lesson is about continuity and computing limits. Continuity is the foundation for optimization and numerical techniques which will come later. It is one key to computing limits. The other key is algebra. This lesson provides an opportunity to hone our algebraic skills while no one is looking.

What's Here 1. Warming Up The Definition of Continuity Discontinuities Removable Discontinuites Jump Discontinuities Other Essential Discontinuities 2. The Field Guide to Continuity Polynomials, Exponential, Logarithmic, and Power Functions Rational Functions Trigonometric functions Piecewise functions Compositions 3. Limit Laws Algebraic Laws The Algebra of Infinity The Squeeze Law Examples Involving sin(x)/x An Exponential Limit Limit Laws and Continuity 4. Continuity Theorems The Intermediate Value Theorem Inside the IVT The Intermediate Zero Theorem The Extreme Value Theorem

What To Do This lesson is an even mixture of concepts (Sections 1. and 4.) and computational techniques. Pictures, examples, counterexamples, and definitions help understand concepts. Computational techniques require examples and practice. Do what you need to do! Expected Study Time You should expect to spend approximately SIX hours working through this Lesson in order to prepare for the Practice area and "Onward and Upward" activities of this Stage. Objectives: Determine the continuity of a function at a point and on its domain. Apply continuity and limit laws to compute a variety limits, including limits arising as difference quotients, limits involving sin(x)/x as x approaches 0, and arithmetic expressions containing the symbol . Recognize valid applications of the Intermediate Value and Extreme Value Theorems. References: The Field Guide to Functions.