mat 1234 calculus i section 2.3 part i using the limit laws
TRANSCRIPT
MAT 1234Calculus I
Section 2.3 Part I
Using the Limit Laws
http://myhome.spu.edu/lauw
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!!!
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
Preview
Limit LawsDirect Substitution PropertyPractical summary of all the limit laws
Limit Laws
11 limit laws that “help” us to compute limits.
Foundation of computing limits, but tedious to use.
Practical methods will be introduced.
Limit Laws
7. limx ac c
x
y
a
c y c
Limit Laws
8. limx ax a
x
y
a
y x
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
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
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
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
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
Why?
Polynomials are “continuous” functions
x
y
a
lim ( ) ( )x a
f x f a
Why?
Polynomials are “continuous” functionslim ( ) lim ( ) ( )x a x a
f x f x f a
x
y
a
( )f a
Example 1 (Polynomial)
Remark 1
Once you substitute in the number, you do not need the limit sign anymore.
Example 2 (Rational Function, a in the domain)
3 is in the domain of the rational function
2
3
6lim
5x
x
x
Example 2 (Rational Function, a in the domain)
2
3
6lim
5x
x
x
3 is in the domain of the rational function
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.
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,
Example 3
3 3 2
1lim 8x
x x
Q&A
Q: What to do if the answer is undefined when plugging in x=a?
A: Try the following techniques
Example 4 (Simplify)
2
1
1lim
1x
x
x
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
Reminders
4. Do not actually “cross out” terms.
1
1limx
x
1
1
x
x
Remark 1 Again
Once you substitute in the number, you do not need the limit sign anymore.
1
lim 1
1 1x
x
Example 5 (Combine the terms)
21
1 2lim
1 1x x x
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
Example 7 (Multiply by conjugate)
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
Example 7 (Multiply by conjugate)
0
2 2limh
h
h
2 2
a b a b
Review: We learned…
Limit Laws Direct Substitution Property of
polynomials and rational functions Techniques
• Simplify
• Combine the terms
• Multiply by conjugate
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.