1.2:Rates of Change & Limits Learning Goals: © 2009 Mark Pickering Calculate average & instantaneous speed Define, calculate & apply properties of limits Use Sandwich Theorem

Important Ideas Limits are what make calculus different from algebra and trigonometry Limits are fundamental to the study of calculus Limits are related to rate of change Rate of change is important in engineering & technology

Theorem 1 Limits have the following properties: if& then: 1.

Theorem 1 Limits have the following properties: if& then: 2.

Theorem 1 Limits have the following properties: if& then: 3.

Theorem 1 Limits have the following properties: if then: 4. & k a constant

Theorem 1 Limits have the following properties: if& then: 5.

Theorem 1 Limits have the following properties: if& 6. r & s are integers, then:

Theorem 1 Limits have the following properties: if where k is a 7. constant, then: (not in your text as Th. 1)

Theorem 2 For polynomial and rational functions: a. b. Limits may be found by substitution

Example Solve using limit properties and substitution:

Try This Solve using limit properties and substitution: 6

Example Sometimes limits do not exist. Consider: If substitution gives a constant divided by 0, the limit does not exist (DNE)

Example Trig functions may have limits.

Try This

Example Find the limit if it exists: Try substitution

Example Find the limit if it exists: Substitution doesn’t work…does this mean the limit doesn’t exist?

Important Idea and are the same except at x =-1

Important Idea The functions have the same limit as x  -1

Procedure 1.Try substitution 2. Factor and cancel if substitution doesn’t work 3.Try substitution again The factor & cancellation technique

Try This Find the limit if it exists: 5 Isn’t that easy? Did you think calculus was going to be difficult?

Try This Find the limit if it exists:

Try This Find the limit if it exists: The limit doesn’t exist Confirm by graphing

Definition When substitution results in a 0/0 fraction, the result is called an indeterminate form.

Important Idea The limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.

Try This Find the limit if it exists: -5

Try This Graph and on the same axes. What is the difference between these graphs?

Why is there a “hole” in the graph at x =1? Analysis

Example Consider for and for x =1 =?

Try This Find: if

Important Idea The existence or non- existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.

Important Idea What matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.

Try This Find: f(0) is undefined; 2 is the limit 2

Find: Try This f(0) is defined; 2 is the limit 2 1

Try This Find the limit of f(x) as x approaches 3 where f is defined by:

Try This Graph and find the limit (if it exists): DNE

Theorem 3: One-sided & Two Sided limits if (limit from right) and (limit from left) then (overall limit)

Theorem 3: One-sided & Two Sided limits (Converse) if (limit from right) and (limit from left) then(DNE)

Example Consider What happens at x =1? x f(x) Let x get close to 1 from the left:

Try This Consider x f(x) Let x get close to 1 from the right:

Try This What number does f(x) approach as x approaches 1 from the left and from the right?

Try This Find the limit if it exists: DNE

Example Find the limit if it exists:

Example 1.Graph using a friendly window: 2. Zoom at x =0 3. Wassup at x =0?

Important Idea If f(x) bounces from one value to another (oscillates) as x approaches c, the limit of f(x) does not exist at c :

Theorem 4: Sandwich (Squeeze) Theorem Let f(x) be between g(x) & h(x) in an interval containing c. If then: f(x) is “squeezed” to L

Example Find the limit if it exists: Where  is in radians and in the interval

Example Find the limit if it exists: Substitution gives the indeterminate form…

Example Find the limit if it exists: Factor and cancel doesn’t work…

Example Find the limit if it exists: Maybe…the squeeze theorem…

Example g(  )=1 h(  )= cos 

Example & therefore…

Two Special Trig Limits Memorize

Example Find the limit if it exists:

Example Find the limit if it exists:

Try This Find the limit if it exists: 0

Lesson Close Name 3 ways a limit may fail to exist.

Practice 1. Sec 1.2 #1, 3, 8, 9-18, 28-38E (just find limit L), 39-42gc (graphing calculator), 43-45