MAT 1234 Calculus I Section 2.3 Part I Using the Limit Laws .
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Transcript of MAT 1234 Calculus I Section 2.3 Part I Using the Limit Laws .
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MAT 1234Calculus I
Section 2.3 Part I
Using the Limit Laws
http://myhome.spu.edu/lauw
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Quiz Tomorrow and …
Quiz :1.5, 1.6I Homework 1.6 Part I Do your HW ASAP. Write out your solutions carefully in a
notebook - You want to have a reference before the exams…and bonus points for your first exam
Tutoring is available!!!
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Recall
Limit of the following form is important
1.4: Estimate limits by tables 1.6: Compute limits by algebra 1.5: Formally define limits
h
afhafh
)()(lim
0
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Preview
Limit LawsDirect Substitution PropertyPractical summary of all the limit laws
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Limit Laws
11 limit laws that “help” us to compute limits.
Foundation of computing limits, but tedious to use.
Practical methods will be introduced.
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Limit Laws
7. limx ac c
x
y
a
c y c
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Limit Laws
8. limx ax a
x
y
a
y x
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Limit Laws
If and exist, then )(lim xfax
)(lim xgax
1. lim ( ) ( ) lim ( ) lim ( )
3. lim ( ) lim ( )x a x a x a
x a x a
f x g x f x g x
cf x c f x
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Example 1
1. lim ( ) ( ) lim ( ) lim ( )
3. lim ( ) lim ( )
7. lim
8. lim
x a x a x a
x a x a
x a
x a
f x g x f x g x
cf x c f x
c c
x a
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Direct Substitution Property
If f(x) is a polynomial, then
Also true if f(x) is a rational function and a is in the domain of f
)()(lim afxfax
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Direct Substitution Property
If f(x) is a polynomial, then
Also true if f(x) is a rational function and a is in the domain of f
)()(lim afxfax
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Direct Substitution Property
If f(x) is a polynomial, then
Also true if f(x) is a rational function and a is in the domain of f
)()(lim afxfax
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Why?
Polynomials are “continuous” functions
x
y
a
lim ( ) ( )x a
f x f a
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Why?
Polynomials are “continuous” functionslim ( ) lim ( ) ( )x a x a
f x f x f a
x
y
a
( )f a
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Example 1 (Polynomial)
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Remark 1
Once you substitute in the number, you do not need the limit sign anymore.
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Example 2 (Rational Function, a in the domain)
3 is in the domain of the rational function
2
3
6lim
5x
x
x
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Example 2 (Rational Function, a in the domain)
2
3
6lim
5x
x
x
3 is in the domain of the rational function
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Direct Substitution Property
Can be extended to other functions such as n-th root.
Not for all functions such as absolute value, piecewise defined functions.
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Limit Laws Summary
Use Direct Substitutions if possible*. That is, plug in x=a when it is defined.
)(lim xfax
* Sums, differences, products, quotients, n-th root functions of polynomials,
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Example 3
3 3 2
1lim 8x
x x
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Q&A
Q: What to do if the answer is undefined when plugging in x=a?
A: Try the following techniques
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Example 4 (Simplify)
2
1
1lim
1x
x
x
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1.Use equal signs
2.Use parentheses for expressions with sums and differences of more than 1 term.
3. Show the substitution step.
Reminders
1
lim 1x
x
1
lim 1
1 1x
x
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Reminders
4. Do not actually “cross out” terms.
1
1limx
x
1
1
x
x
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Remark 1 Again
Once you substitute in the number, you do not need the limit sign anymore.
1
lim 1
1 1x
x
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Example 5 (Combine the terms)
21
1 2lim
1 1x x x
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Remark 1 Again (What? Again!)
Once you substitute in the number, you do not need the limit sign anymore.
1
1lim
11
1 1
x x
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Example 7 (Multiply by conjugate)
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Review of conjugates
The conjugate of is
The conjugate of is
The product of conjugates is
ba ba
ba ba
2 2
a b
a b
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Example 7 (Multiply by conjugate)
0
2 2limh
h
h
2 2
a b a b
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Review: We learned…
Limit Laws Direct Substitution Property of
polynomials and rational functions Techniques
• Simplify
• Combine the terms
• Multiply by conjugate
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Classwork
Use pencils Use “=“ signs Do not “cross out” anything. Do not skip steps
Once you substitute in the number, you do not need the limit sign anymore.