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![Page 1: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/1.jpg)
Definition and finding the limit
When substitution results in a 0/0 fraction, the result is called an indeterminate form.
The limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.
![Page 2: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/2.jpg)
Calculus Date: 9/26/14
Objective: SWBAT define, calculate & apply properties of limits graphically, numerically and now analytically.
Do Now –
Mini Quiz 5 minutes Take out a piece of paper. Can be a half sheet.
HW Requests: HW: pg 30 SM all
In class: worksheets ( possibilities Worksheet limits or McCafrey)
HW: Complete WorksheetsAnnouncements:
Mandatory session Sine and Cosine functions starting
with the Unit Circle
Quiz Friday
To get ahead,You have to do extra!
![Page 3: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/3.jpg)
Mini Quiz7 minutes
2.Show your work then
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1. Try Substitution, if doesn’t work
2. Try Factor and cancel and then3. Try Substitution again, if doesn’t work4. Do DNE or +/- infinity check
Techniques-Finding limits for Rational Expressions
Let’s go to the SM pg #28 #1-8 (10) HW: pg 30 SM all
- If the right and left side limit are not equal the limit does not exist - DNE
![Page 5: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/5.jpg)
- If the right and left side limit are not equal the limit does not exist – DNE
One sided Limits- If the left side number is negative then the - If the left side number is positive then the - If the right side number is negative then the - If the left side number is negative then the
![Page 6: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/6.jpg)
Rationalizing Technique
x
xx
11lim
0
If there is a radical in the numerator or the denominator,rationalize, simplify and cancel, then try substitution.
)11(
1
)11()11(
11
11
11.
11
x
xx
x
xx
x
x
x
x
x
Substituting we get 2
1
)110(
1
Hint: Often you can cancel a common term in the numerator and denominator when simplifying
![Page 7: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/7.jpg)
Rationalizing Technique
xxx
x
x
x
x
x
2
1
)2)(4(
4
2
2.
4
2
Rationalize, simplify (cancel) and try substitution.
Substituting we get = 1 4
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Try This
Find:
f(0)is undefined; 2 is the limit
2( )
1 1
xf x
x
0lim ( )x
f x
![Page 9: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/9.jpg)
Find: Try This
( ) , 01 1
xf x x
x
f(0) is defined; 2 is the limit
21
0lim ( )x
f x
1, x = 0
![Page 10: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/10.jpg)
Try ThisFind the limit if it exists:
0limx
x
x
DNE
![Page 11: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/11.jpg)
Try ThisFind: if
1lim ( )x
f x
1lim ( ) 3x
f x
![Page 12: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/12.jpg)
Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:
2 , 3( )
3 , 3
xf x
x
3lim ( ) 2x
f x
![Page 13: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/13.jpg)
Example
Find the limit if it exists:3
1
1lim
1x
x
x
Try substitution
![Page 14: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/14.jpg)
Example
Find the limit if it exists:3
1
1lim
1x
x
x
Substitution doesn’t work…does this mean the limit doesn’t exist?
![Page 15: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/15.jpg)
Use the factor & cancellation technique
3 21 ( 1)( 1)
1 1
x x x x
x x
2 1x x and
are the same except at x=-1
![Page 16: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/16.jpg)
Use the factor & cancellation technique
2 1x x
After factoring and cancelling, now try substituting -1 again.
= 33
1
1lim
1x
x
x
![Page 17: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/17.jpg)
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
5
Isn’t
that
easy?
Did you think ca
lculus
was going to
be
difficu
lt?
![Page 18: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/18.jpg)
Try ThisSolve using limit properties and substitution:2
2
4lim
3x
x x
x
6
![Page 19: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/19.jpg)
Try ThisFind the limit if it exists:
22
2lim
4x
x
x
1
4
![Page 20: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/20.jpg)
ExampleSometimes limits do not exist. Consider:
3
2
3lim
2x
x
x
If substitution gives a constant divided by 0, the limit does not exist (DNE)
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Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
The limit doesn’t existConfirm by graphing
![Page 22: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/22.jpg)
Lesson Close
Name 3 ways a limit may fail to exist.
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Exit Ticket
• In Class: SM – pg 28 #1-5
• HW: SM pg 30 #1-15
![Page 24: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/24.jpg)
Try ThisFind the limit if it exists:
2
1
2 3lim
1x
x x
x
-5
![Page 25: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/25.jpg)
Limit properties again
The 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.
What matters is…what value does f(x) get very, very close to as x gets very, very close to c. This value is the limit.
![Page 26: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/26.jpg)
In order for a limit to exist at c, the left-hand limit must equal the right hand limit.
Limits, again!
)(lim)(lim xfxfcxcx
Lxfcx
)(lim
If the left-hand limit equals the right hand limit, then the limit exists and we write:
Watch out for piecewise functions
![Page 27: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/27.jpg)
When finding the limit of a function it is important to let x approach a from both the right
and left. If the same value of L is approached by the function then
the limit exist and
Lxf )(limax
![Page 28: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/28.jpg)
ExampleConsider3 1
( )1
xf x
x
for ( ,1) (1, ) and
3
1
1lim
1x
x
x
=?
![Page 29: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/29.jpg)
Try ThisGraph and find the limit (if it exists):
3
3lim
3x x DNE
![Page 30: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/30.jpg)
ExampleTrig functions may have limits.
2
lim(sin )x
x
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Try This
2
lim(cos )x
x
2
lim(cos ) cos 02x
x
![Page 32: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/32.jpg)
Using the Product Rule Technique
![Page 33: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/33.jpg)
Important Idea
The functions have the same limit as x-1
![Page 34: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/34.jpg)
Try This
Graph and
3
1
1
1
xY
x
2
2 1Y x x on the same axes. What is the
difference between these graphs?
![Page 35: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/35.jpg)
3 1( )
1
xf x
x
Why is there a “hole” in the graph at x=1?
Analysis
![Page 36: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/36.jpg)
Example
Consider
3 1( ) , 1
1
xf x x
x
What happens at x=1?
x .75 .9 .99 .999
f(x)
Let x get close to 1 from the left:
![Page 37: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/37.jpg)
Try This
Consider
3 1( ) , 1
1
xf x x
x
x 1.25 1.1 1.01
1.001
f(x)
Let x get close to 1 from the right:
![Page 38: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/38.jpg)
Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?
3
1
1lim 3
1x
x
x
![Page 39: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/39.jpg)
Informal Definition of Limit• If f(x) becomes arbitrarily close to a single
number L as x approaches a number c from either side, the limit of f(x), as x approaches c, is L.
Lxfcx
lim
![Page 40: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/40.jpg)
- Definition of Limit
• Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement means that for each >0 there exists a >0 such that if then .
Lxfcx
lim
cx0 Lxf
![Page 41: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/41.jpg)
Basic Limits
•
•
•
bbcx
lim
cxcx
lim
nn
cxcx
lim
![Page 42: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/42.jpg)
Constant Function Limits
bbax
lim
• a and b are both constants
• This means that for any constant function f(x) = b, as x approaches any constant a, the limit will always be b.
![Page 43: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/43.jpg)
Linear Function Limits
axax
lim
The limit of f(x) = x as x approaches any constant is the constant itself.
![Page 44: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/44.jpg)
Exponential Function Limits
ax
axbb
lim
Just plug in a for x
![Page 45: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/45.jpg)
Properties
Let and
Scalar multiple: Sum or difference:
Product:
Quotient: , if K0
Power:
Lxfcx
lim Kxgcx
lim
bLxbfcx
lim
KLxgxfcx
lim
LKxgxfcx
lim
K
L
xg
xfcx
lim
nn
cxLxf
lim
![Page 46: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/46.jpg)
• Let's try a practice problem.
• Property (B) tells us we can split these apart:
• Using limit (1) and limit (2) from the basic limits, we get:
3lim2
xx
3limlim22
xx
x
1323lim
33lim
2lim
2
2
2
x
x
x
x
x
![Page 47: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/47.jpg)
Putting it all together
• So,
5
714131
743lim
23
23
1
xxxx
This is called the Substitution method
![Page 48: Definition and finding the limit When substitution results in a 0/0 fraction, the result is called an indeterminate form. The limit of an indeterminate.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e495503460f94b3caae/html5/thumbnails/48.jpg)
Try ThisSolve using limit properties and substitution:
2
3lim 2 3 2x
x x
25