Finding Limits Graphically and Numerically

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Finding Limits Graphically and Numerically

• Lesson 2.2

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Average Velocity

• Average velocity is the distance traveled divided
  by an elapsed time.

A boy rolls down a hill on a skateboard. At time
4 seconds, the boy has rolled 6 meters from the
top of the hill. At time 7 seconds, the boy
has rolled to a distance of 30 meters. What is
his average velocity?
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Distance Traveled by an Object

• Given distance s(t) 16t2
• We seek the velocity
• or the rate of change of distance
• The average velocity between 2 and t

2 t
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Average Velocity

• Use calculator
• Graph with window 0 lt x lt 5, 0 lt y lt 100
• Trace for x 1, 3, 1.5, 1.9, 2.1, and then x
  2
• What happened?

This is the average velocity function
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Limit of the Function

• Try entering in the expression
  limit(y1(x),x,2)
• The function did not exist at x 2
• but it approaches 64 as a limit

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Limit of the Function

• Note we can approach a limit from
• left right both sides
• Function may or may not exist at that point
• At a
• right hand limit, no left
• function not defined
• At b
• left handed limit, no right
• function defined

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Observing a Limit

• Can be observed on a graph.

ViewDemo
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Observing a Limit

• Can be observed on a graph.

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Observing a Limit

• Can be observed in a table
• The limit is observed to be 64

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Non Existent Limits

• Limits may not exist at a specific point for a
  function
• Set
• Consider the function as it approaches x 0
• Try the tables with start at 0.03, dt 0.01
• What results do you note?

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Non Existent Limits

• Note that f(x) does NOT get closer to a
  particular value
• it grows without bound
• There is NO LIMIT
• Try command oncalculator

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Non Existent Limits

• f(x) grows without bound

View Demo3
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Non Existent Limits
View Demo 4
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Formal Definition of a Limit

• The
• For any e (as close asyou want to get to L)
• There exists a ? (we can get as close as
  necessary to c )

View Geogebra demo
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Formal Definition of a Limit

• For any ? (as close as you want to get to L)
• There exists a ? (we can get as close as
  necessary to cSuch that

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Specified Epsilon, Required Delta
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Finding the Required ?

• Consider showing
• f(x) L 2x 7 1 2x 8 lt ?
• We seek a ? such that when x 4 lt ?
• 2x 8lt ? for any ? we choose
• It can be seen that the ? we need is

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Assignment

• Lesson 2.2
• Page 76
• Exercises 1 35 odd