Chapter 13: Limit and Derivative

Introduction

• Importance: Understanding the concept of limits is very important.
• Objective: To lay the foundation of limits and derivatives.

Basic Concepts of Limit

• What is a Limit?
  • Study of variation in value in calculus.
  • Studying the variation in the value of a function as the variable changes in the domain.
• Functional Value: Can determine the functional value for all values in the domain.
• Indeterminate Forms: Situations where the functional value cannot be determined.

Importance of Limit

• Limiting Value: The value obtained from functional points.
• Left and Right Hand Limits:
  • Left Hand Limit: When x approaches a from the left.
  • Right Hand Limit: When x approaches a from the right.

Indeterminate and Undefined Forms

• Undefined Form: Some values are infinite, like x/0.
• Indeterminate Forms: Situations like 0/0 which are indicated but not clear.
• There are 7 types of indeterminate forms.

Deleted Neighborhood

• Neighborhood and Deleted Neighborhood:
  • Neighborhood: Points around a point.
  • Deleted Neighborhood: Removing the main point from the neighborhood.

Examples

1. f(c) = x² - 4 / x - 2
   • Extracting values from the deleted neighborhood.
   • Left and Right Hand Limits:
     • In both situations, the limit is 4.
2. Examples of different limiting values.

Conclusion

• Importance of Limit and Functional Value:
  • Both are not always the same.
• Existence of Limit: When left and right limits are equal.
• Errors and Solutions: Emphasis on understanding and applying limits correctly.

Notes and Suggestions

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