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Section 1.3: Evaluating Limits Analytically. Example. Let and . Find:. Direct Substitution. One of the easiest and most useful ways to evaluate a limit analytically is direct substitution ( substitution and evaluation): - PowerPoint PPT Presentation
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Section 1.3: Evaluating Limits Analytically
Example
Let and . Find: 4 2f 4 5g 2
3 5 7
2
3 5 7
4
15 7
4
8
4
2
2
3 4 7
4
g
f
Direct SubstitutionOne of the easiest and most useful ways to evaluate a limit analytically is direct substitution (substitution and evaluation):
If you can plug c into f(x) and generate a real number answer in the range of f(x), that generally implies that the limit exists (assuming f(x) is continuous at c).
Example: 3
2limxx
32 8
Always check for substitution first.
The slides that follow investigate why Direct Substitution is valid.
Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
Constant Function
Limit of x
Limit of a Power of x
Scalar Multiple
lim ( )x cf x L
lim ( )
x cg x K
limx cb b
limx cx c
lim n n
x cx c
lim ( )x c
b f x b L
Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
Sum Difference
Product
Quotient
Power
lim ( )x cf x L
lim ( )
x cg x K
lim ( ) ( )x c
f x g x L K
lim ( ) ( )x c
f x g x L K
( )lim , 0
( )x c
f x LK
g x K
lim ( )n n
x cf x L
ExampleLet and . Find the following limits.
5lim 4xf x
5lim 2xg x
1. limx 5
f x 5g x
limx 5f x +lim
x 5 5g x
limx 5
f x +5limx 5
g x
2. limx 5
f x g x
3. limx 5
f x g x
4 +5 2
6
limx 5f x lim
x 5 g x
4 2
8
limx 5
f x
limx 5
g x
4 2
2
Example 2
5 3 2
2Evaluate lim 2 9 3 11
xx x x
5 3 2
2 2 2 2lim 2 lim 9 lim 3 lim 11x x x x
x x x
Sum/Difference Property
5 3 2
2 2 22 lim 9 lim 3 lim 11
x x xx x x
Multiple and Constant Properties5 3 2
2 2 22 lim 9 lim 3 lim 11
x x xx x x
Power Property
5 3 22 2 9 2 3 2 11
Limit of x Property
7
Dire
ct S
ubst
itutio
n
Direct substitution is a valid analytical method to evaluate the following limits.
• If p is a polynomial function and c is a real number, then:
• If r is a rational function given by r(x) = p(x)/q(x), and c is a real number, then
• If a radical function where n is a positive integer. The following limit is valid for all c if n is odd and only c>0 when n is even:
Direct Substitution
lim ( ) ( )x cp x p c
( )lim ( ) ( ) , ( ) 0
( )x c
p cr x r c q c
q c
lim n n
x cx c
Direct substitution is a valid analytical method to evaluate the following limits.
• If the f and g are functions such that Then the limit of the composition is:
• If c is a real number in the domain of a trigonometric function then:
Direct Substitution
lim ( ) lim ( ) ( )x c x Lg x L and f x f L
lim ( ( )) lim ( ) ( )x c x cf g x f g x f L
limsin sinx c
x c
lim cos cosx c
x c
lim tan tanx c
x c
lim cot cotx c
x c
limsec secx c
x c
lim csc cscx c
x c
Example
3
2Evaluate lim
6x
x
x
3 2
3 6
Direct Substitution can be used since the
function is well defined at x=3
1
9
For what value(s) of x can the limit not be evaluated using direct substitution?
At x=-6 since it makes the denominator 0: 6 6 0
Indeterminate Form
0
0
An example of an indeterminate form because the limit can
not be determined. 1/0 is another example.
Often limits can not be evaluated at a value using Direct Substitution. If this is the case, try to find another function that agrees with the original function except at the point in question. In other words…
How can we simplify: ?2
24 4
2x xx x
Evaluate the limit analytically:2
24 4
22lim x x
x xx
2
2
2 4 2 4
2 2 2
Strategies for Finding LimitsTo find limits analytically, try the following:
1. Direct Substitution (Try this FIRST)
2. If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include…
• Factoring/Dividing Out Technique• Rationalize Numerator/Denominator• Eliminating Embedded Denominators • Trigonometric Identities• Legal Creativity
Example 1Evaluate the limit analytically:
2
24 4
22lim x x
x xx
2 2
2 12lim x x
x xx
Factor the numerator and denominator
2 2
2 12lim x x
x xx
Cancel common factors
212
lim xxx
2 22 1 Direct substitution
0
At first Direct Substitution fails
because x=2 results in dividing by zero
This function is equivalent to the
original function except at x=2
Example 2Evaluate the limit analytically:
2 222
lim yyy
2 2
2 2
y
y
Rationalize the numerator
2 4
2 2 22lim y
y yy
Cancel common factors
2
2 2 22lim y
y yy
12 22
limyy
Direct substitution1 1
42 2 2
Example 3Evaluate the limit analytically:
1 13
33lim x
xx
33xx
Cancel the denominators of the fractions
in the numerator
3
3 33lim x
x xx
If the subtraction is
backwards, Factoring a negative 1 to flip the signs
3
3 33lim x
x xx
3
3 33lim x
x xx
Direct substitution1 13 3 9
Cancel common factors
133
lim xx
limh 0
h h 10 h
Example 4Evaluate the limit analytically:
Expand the the expression
to see if anything cancels
Direct substitution
Factor to see if anything cancels
limh 0
h 5 2 25h
limh 0
h 5 h 5 25h
limh 0
h 2 10h25 25h
limh 0
h 2 10hh
10
limh 0
h 10
0 10
limx 4
sin x cosx sin x cosx cosx
Example 5Evaluate the limit analytically:
Rewrite the tangent
function using cosine and
sine
Direct substitution
Eliminate the embedded fraction
limx 4
1 tan xsin x cosx
limx 4
1 sin xcos x
sin x cosx
cosxcosx
limx 4
cosx sin xsin x cosx cosx
12 2
limx 4
1cosx
1cos
4
If the subtraction is backwards,
Factoring a negative 1 to flip
the signs
2
Two “Freebie” Limits
0
sinlim 1x
x
x
0
1 coslim 0x
x
x
The following limits can be assumed to be true (they will be proven later in the year) to assist in finding other limits:
Use the identities to help with these limits. They are located on the first page of your textbook.
Example Evaluate the limit analytically:
sin350
lim xxx
33
If 3x is the input of the
sine function then 3x needs
to be in the denominator
3sin35 30
lim xxx
3 sin35 30
lim xxx
3 sin35 30
lim xxx
Assumed Trig Limit
35
Scalar Multiple Property
35 1
Isolate the “freebie”
limx 0
sin 2 xx sin x 1cosx
ExampleEvaluate the limit analytically:
Try multiplying by the
reciprocal
A freebie limit and Direct substitution
limx 0
1 cosxx sin x
limx 0
1cosx cosx cos2 xx sin x 1cosx
1cosx1cosx
limx 0
1 cos2 xx sin x 1cosx
limx 0
sin xx 1cosx
1 11cos0
Use the Trigonometry
Laws
12
limx 0
sin xx lim
x 0
11cosx
Split up the limits
