Calculus 2.1 day 1

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The Limit of a Function
Grand Teton National Park, Wyoming
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Suppose you drive 200 miles, and it takes you 4
hours.
If you look at your speedometer during this trip,
it might read 65 mph. This is your instantaneous
speed.
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A rock falls from a high cliff.
The position of the rock is given by
After 2 seconds
average speed
What is the instantaneous speed at 2 seconds?
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for some very small change in t
where h some very small change in t
We can use the TI-89 to evaluate this expression
for smaller and smaller values of h.
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We can see that the velocity approaches 64 ft/sec
as h becomes very small.
We say that the velocity has a limiting value of
64 as h approaches zero.
(Note that h never actually becomes zero.)
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The limit as h approaches zero
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Consider
What happens as x approaches zero?
Graphically
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(No Transcript)
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Numerically
You can scroll down to see more values.
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You can scroll down to see more values.
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Limit notation
So
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not 1
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Properties of Limits
Limits can be added, subtracted, multiplied,
multiplied by a constant, divided, and raised to
a power.
(See your book for details.)
For a limit to exist, the function must approach
the same value from both sides.
One-sided limits approach from either the left or
right side only.
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2
1
1
2
3
4
At x1
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2
1
1
2
3
4
At x2
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2
1
1
2
3
4
At x3
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The Sandwich Theorem
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By the sandwich theorem
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p