1.2:Rates of Change & LimitsLearning Goals:2009 Mark Pickering Calculate average & instantaneous speedDefine, calculate & apply properties of limitsUse Sandwich Theorem

Important IdeasLimits are what make calculus different from algebra and trigonometryLimits are fundamental to the study of calculusLimits are related to rate of changeRate of change is important in engineering & technology

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

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

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

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

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

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

Theorem 1Limits have the following properties:ifwhere k is a7.constant, then:(not in your text as Th. 1)

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

ExampleSolve using limit properties and substitution:

Try ThisSolve using limit properties and substitution:6

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

ExampleTrig functions may have limits.

Try This

ExampleFind the limit if it exists:Try substitution

ExampleFind the limit if it exists:Substitution doesnt workdoes this mean the limit doesnt exist?

Important Ideaandare the same except at x=-1

Important IdeaThe functions have the same limit as x-1

ProcedureTry substitution Factor and cancel if substitution doesnt workTry substitution againThe factor & cancellation technique

Try ThisFind the limit if it exists:5Isnt that easy?Did you think calculus was going to be difficult?

Try ThisFind the limit if it exists:

Try ThisFind the limit if it exists:The limit doesnt existConfirm by graphing

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

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

Try ThisFind the limit if it exists:-5

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

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

ExampleConsiderforandfor x=1

Try ThisFind: if

Important IdeaThe 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 IdeaWhat matters iswhat value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.

Try ThisFind:f(0)is undefined; 2 is the limit2

Try ThisFind:

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

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

Theorem 3: One-sided & Two Sided limitsif(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)

ExampleConsider What happens at x=1?Let x get close to 1 from the left:

x.75.9.99.999 f(x)

Try ThisConsider Let x get close to 1 from the right:

x1.251.11.011.001 f(x)

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

Try ThisFind the limit if it exists:DNE

ExampleFind the limit if it exists:

Example1.Graph using a friendly window:2. Zoom at x=03. Wassup at x=0?

Important IdeaIf 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) TheoremLet f(x) be between g(x) & h(x) in an interval containing c. Ifthen:f(x) is squeezed to L

ExampleFind the limit if it exists:Where is in radians and in the interval

ExampleFind the limit if it exists:Substitution gives the indeterminate form

ExampleFind the limit if it exists:Factor and cancel doesnt work

ExampleFind the limit if it exists:Maybethe squeeze theorem

Exampleg()=1h()=cos

Example&therefore

Two Special Trig LimitsMemorize

ExampleFind the limit if it exists:

ExampleFind the limit if it exists:

Try ThisFind the limit if it exists:0

Lesson CloseName 3 ways a limit may fail to exist.

Practice1. Sec 1.2 #1, 3, 8, 9-18, 28-38E (just find limit L), 39-42gc (graphing calculator), 43-45