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
To have a limit L as x approaches a, a function f must be defined on both sides of a and its values f(x) must approach L as x approaches a from either sides.
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
Definition: Continuous Function
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
