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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
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":
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.
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.
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 Continuity Definition of the continuity of functions One sided continuity Properties of continuous functions Rate of change