Download Calculus Fall 2010 Lesson 05 _Evaluating limits of

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Lesson Plan #5
Date: Wednesday September 16th, 2009
Class: Intuitive Calculus
Topic: Evaluating Limits
Aim: How do we evaluate limits by rationalizing either the numerator or denominator of a function?
Objectives:
1) Students will be able to evaluate limits by rationalizing either the numerator or denominator of a function.
2) Students will be able to evaluate limits of functions that are expressed as complex fractions, by simplifying the complex fraction
into a simple fraction.
3) Students will be able to evaluate one-sided limits.
HW# 5:
Find the limit if it exists
x7 3
x
1) lim
x 0
1
1
x

2
2) lim
x  1
x 1
3) lim  f ( x ) , where graph of f (x ) is
x  1
Do Now:
1)
2)
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
In the following limit, what happens when we attempt to evaluate the limit using direct substitution?
lim
x 0
x 1 1
x
To evaluate a limit like the one illustrated above, we can try to rationalize the part of the fraction that has a radical.
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Assignment #1:
Rationalize the numerator of function listed above. Then evaluate the limit of this function (which agrees with the previous
function in all but one point so we could use this function to evaluate the limit of the original function)
lim
x 0
x 1 1
……
x
Examples or Exercises (depending on how the lesson is going):
Find the limit (if it exists)
1) lim
3 x  3
x
2) lim
x 1  2
x 3
x 0
x 3
4  18  x
x2
x2
3) lim
Assignment #2:
1
1
In the limit, lim x  1
, what happens when we attempt to evaluate the limit by direct substitution?
x 0
x
What technique should we use to evaluate the above limit?
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Examples or Exercises (depending on how the lesson is going):
Find the limit (if it exists)
1
1

1) lim 5  x 5
x 0
x
1
1

2) lim x  4 4
x 0
x
1
1

3) lim 2  x 2
x 0
x
4)
lim
x

2
1  sin x
cos x
Sample Test Questions:
3
3
x
1) Evaluate lim
x1 x  1
A) -3
B) -1
C) 1
D) 3
E) None of the other choices
Assignment #3:
Use your graphing calculator to sketch the graph of
f ( x)  3  x
What is the highest value of x that can be used for this
function?
If we were going to find
lim 3  x , we would only need to
x 3
check from one side, the left side, since there is nothing on the
right side of three. We could express this as
lim
x 3
3  x . This is known as a one-sided limit.
One-sided limits:
When we talk about limit from the right, we mean that x approaches
c from values greater than c , We denote this by
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lim f ( x)  L
x c
Similarly, the limit from the left means that x approaches
c from values less than c . We denote this by
lim f ( x)  L
x c 
Evaluate the limit
lim
x 3
3 x
Examples or Exercises (depending on how the lesson is going):
Find the limit (if it exists)
1)
2)
3)
lim
x 5
x 2  25
lim
2 x
x2  4
lim
x 2
x4
x 5
x2
x 4
4)
lim f ( x )
x 3
f (x) 
x2
,x 3
2
12  2 x
,x 3
3
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5) Suppose y  f (x) is defined as follows:
f (x) 
x  1,
( 2  x  0 )
2,
( x  0)
 x,
( 0  x  2)
0,
x2
x  4,
( 2  x  4)
Evaluate:
1)
3)
lim f ( x )
2)
lim f ( x )
4)
x  2
x2
lim f ( x)
x0
lim f ( x )
x4
x represents the greatest integer function. The greatest integer function (or floor function) will round any
number down to the nearest integer. The graph of the greatest integer functions looks like this:
The TI-84 designate this function by using
f(x)=int(x) and is found in the MATH NUM menu
Assignment #4:
A) If f ( x)  x , evaluate

i) f (0.7) =
ii) f (1.8) =
iii) f (e) =
B) Evaluate
lim f ( x ) =
i)
x.6
iv) lim f ( x )
x1
ii) lim f ( x ) =
x1
iii) lim f ( x ) =
x1