Rates of Change and Limits AP CALCULUS AB Chapter 2: Limits and Continuity Section 2.1: Rates of Change and Limits

What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

Average and Instantaneous Speed

Section 2.1 – Rates of Change and Limits Instantaneous Speed: To find the instantaneous speed, we start by calculating the average speed over an interval from time t1 to any slight time later t2=t1+h To find the instantaneous speed, we take increasingly smaller values of h --- in other words, we let h approach 0. Then

Definition of Limit (If the horizontal distance between x and c is less than , then the vertical distance between and L is less than ). or as x gets increasingly closer to c, then gets increasingly closer to L.

Definition of Limit continued

Definition of Limit continued

Properties of Limits

Properties of Limits continued Product Rule: Constant Multiple Rule:

Properties of Limits continued

Example Properties of Limits

Polynomial and Rational Functions

Polynomial and Rational Functions

Example Limits

Section 2.1 – Rates of Change and Limits Techniques for Finding Limits Numerically – plug in values that approach c from both the right and left. Algebraically – factor and cancel/simplify first. Then plug in c for x. Graphically. In “well-behaved” functions we can find the by direct substitution of c for x:

Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

Section 2.1 – Rates of Change and Limits Limits of Trigonometric Functions:

Example Limits

Example Limits [-6,6] by [-10,10]

Section 2.1 – Rates of Change and Limits Functions that agree in all but 1 point

Section 2.1 – Rates of Change and Limits Cancellation Techniques for finding Limits If you have a rational function

Section 2.1 – Rates of Change and Limits Rationalization Techniques for Finding Limits

Section 2.1 – Rates of Change and Limits Special Limits from Trigonometry

Section 2.1 – Rates of Change and Limits Remember:

Section 2.1 – Rates of Change and Limits As you approach 2 from the left, you get closer and closer to 12. As you approach 2 from the right, you get closer and closer to 12. So,

Section 2.1 – Rates of Change and Limits Limit of a Composite Function: If and then

One-Sided and Two-Sided Limits

One-Sided and Two-Sided Limits continued

Section 2.1 – Rates of Change and Limits If As x approaches 2 from the left, f(x) approaches 3. As x approaches 2 from the right, f(x) approaches 3. So,

Example One-Sided and Two-Sided Limits Find the following limits from the given graph. 4 o 1 2 3

Section 2.1 – Rates of Change and Limits Limits that do not exist: As x approaches 0 from the left, f(x) approaches 0. As x approaches 0 from the right, f(x) approaches 1. So, does not exist

Section 2.1 – Rates of Change and Limits More Limits that do not exist: As x approaches , from the left, f(x) goes to As x approaches , from the right, f(x) goes to So does not exist.

Section 2.1 – Rates of Change and Limits More Limits that do not exist: Oscillating behavior Graph on calculator and zoom in about 4 times around x=0.

Sandwich Theorem

Sandwich Theorem

Section 2.1 – Rates of Change and Limits The Sandwich Theorem: If for all In some interval about c, and Then g(x) f(x) h(x)