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Limits and Rates of Change

Topic Review on "Title":

Definition of the limit of f(x):
Let be a function such as is the limit of at the point if and only if for any there is a such that whenever satisfies  Equivalently we say that tends to as tends to or write

Limit Operations:

Let and  be functions.

If and both exists, then

1. exists and

2. exists and

3. if also then exists and

Properties of Limits: 

1) If and then
2) If and there exists a positive number such that for then

3) If and there exists a positive number such that for then

4) If is bounded for all (i.e. there is a such that for all and then

Definition of the continuity of functions:
Let be a function. is said to be continuous at  if exists and

One sided continuity:
Let be a function. is said to be left continuous at if exits and Similarly is said to be right continuous at if

Rapid Study Kit for "Title":
Flash Movie Flash Game Flash Card
Core Concept Tutorial Problem Solving Drill Review Cheat Sheet

"Title" Tutorial Summary :

Limits of functions are used to develop the idea of the evaluation of a limit. One sided limits are introduced with the help of example problems. In this tutorial limits are given in their traditional definition. Limit operations and properties of limits are covered with the help of examples. The idea of a one sided limit is mentioned in some of the examples covered in this tutorial.

Continuity and its definition is mentioned in some of the example problems. The concept of one-sided continuity and the properties of continuous functions is mentioned in one part of this tutorial. The rate of change principle is showing with a graphical and a physical interpretation.

Tutorial Features:

Specific Tutorial Features:

  • Step by step examples are shown to describe the definition of a limit and the continuity principle.
  • The rate of change idea is presented in some of the example problems with the help of a graphical and physical interpretation.

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 and properties of limits
Evaluation of limits
One sided limits
Definition of the continuity of functions
One sided continuity
Properties of continuous functions
Rate of change

See all 24 lessons in college calculus, including concept tutorials, problem drills and cheat sheets:
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