Chapter 1: Limit and Continuity

Chapter 1: Limit and Contunity

Rates of Change and Limits

Average Rate of Change

Instantaneous Rate of Change

Definition of Limit

Situations where limits fail to exist

Jump

Grows too large to have a limit

Oscillates too much to have a limit

Finding Limits and One-Sided Limits

Properties of Limits

Theorem 1: Limit Rules

Theorem 2: Limits of Polynomials Can Be Found by Substitution

Theorem 3: Limits of Rational Functions Can Be Found by Substitution

Eliminate Zero Denominator

Sandwich Theorem

Two-Sided Limit

One-Sided Limit

Right-Hand Limit

Left-Hand Limit

Relation between One-sided and Two-sided limits

Limits Involving Infinity

Symbol: Infinity

Limits that approaches infinity and minus infinity

Rules for limits as x approaches infinity and minus infinity

Asymptotes

Horizontal Asymptote

x approaches (minus) infinity 

Vertical Asymptote

x approaches values where denominator is zero

End Behavior Model

Oblique Asymptote

Continuity

Definition: Continuous Function

Continuity Test

1. f(c) exists

2. lim(x->c) f(x) exists

3. lim(x->c) f(x) = f(c)

Types of discontinuities

Jump discontinuity

Infinite discontinuity

Oscillating discontinuity

Properties of Continuous Function

Composite of Continuous Functions

The Intermediate Value Theorem for Continuous Functions

Tangent Lines

Definition for Tangent Lines

Slope and Tangent Line

Derivative at a point