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Limits and Rates of Change

 Topic Review on "Title": Definition of limits: Let be a function,  is the limit of  at the point if and only if for any , there is a such that whenever satisfies . Limit operations: The sum of the limits of two functions is equal to the sum of the first limit plus the sum of the second limit. The product of the limits of two functions is equal to the first limit times the second limit. The quotient of the limits of two functions is equal to first limit divided by the second limit.  Definition of continuity of functions: Let be a function, is said to be continuous at if exists and .   Derivative Operations: Let be functions. If and are both differentiable at then 1)  is differentiable at and 2)  is differentiable at and 3) If then is differentiable at and .

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 "Title" Tutorial Summary : This tutorial discusses the limits and the rates of change. Knowing the concept of limit process and instantaneous change is important to the formulation of derivatives and approximation of solutions. The different types of limits that one gets are discussed in the graphical illustrations. Properties of limits and limit operations are presented so an evaluation of limits can be performed. An example of how a function can have no limit but be bounded by 1 is discussed. The principle of continuity relies on limits and it has properties that are useful with the rates of change concept.

 Tutorial Features: Specific Tutorial Features: • Graphical illustration of one sided limits and the continuity of functions is shown in the examples. • Typical rate of change problems are introduced with graphical interpretations. Series Features: • Concept map showing inter-connections of new concepts in this tutorial and those previously introduced. • Definition slides introduce terms as they are needed. • Visual representation of concepts • Animated examples—worked out step by step • A concise summary is given at the conclusion of the tutorial.

 "Title" Topic List: `Limit of functions Definition of limits Limit operations Properties of limits One sided limitsContinuity Definition of the continuity of functions One sided continuity Properties of continuous functionsRate of change Derivative operations `

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