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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