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Math Tutoring TBA Please note: You will be assigned “Mandatory Math Tutoring” if you miss 3...
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Transcript of Math Tutoring TBA Please note: You will be assigned “Mandatory Math Tutoring” if you miss 3...
Math TutoringTBAPlease note: You will be assigned “Mandatory Math Tutoring” if you miss 3 assignments
Calculus TutoringTuesday – 3-4 pm rm. 655Thursday – 3-4 pm rm. 680
2.2 (CONT) Limits: 2.2 (CONT) Limits: numerically numerically Graphically estimate the limit of:Graphically estimate the limit of:
2
1lim 3 2x
x x
Estimate the limit using a table of values.Estimate the limit using a table of values.
Limits: numerically Limits: numerically Graphically estimate the limit of:Graphically estimate the limit of:
3
1
1lim
1x
x
x
Estimate the limit using a table of values.Estimate the limit using a table of values.
Limits: numerically Limits: numerically Graphically estimate the limit of:Graphically estimate the limit of:
Estimate the limit using a table of values.Estimate the limit using a table of values.
21
1lim .
1x
xx
Limits: numerically Limits: numerically Graphically estimate the limit of:Graphically estimate the limit of:
Estimate the limit using a table of values.Estimate the limit using a table of values.
2
20
9 3lim .t
tt
COOL FoGCOOL FoG
1.1. 00
2.2. 00
3.3. 11
4.4. 55
5.5. 11
6.6. 11
7.7. 11
8.8. 00
9.9. 22
10.10.11
11.11.11
12.12.11
Section 2.3Section 2.3
Calculating Limits Calculating Limits Algebraically Algebraically
SWBATSWBAT– Calculate limits using limit lawsCalculate limits using limit laws
Direct Substitution PropertyDirect Substitution Property
Let’s start with the Let’s start with the following:following:
If you can evaluate the limit at a, then do it.If you can evaluate the limit at a, then do it.This is also known as “This is also known as “Plug and ChugPlug and Chug””
lim ( ) ( )x a
f x f a
2
5 2x – 3x 4lim
x
If you cant “plug and chug”,If you cant “plug and chug”,Try something algebraic. Try something algebraic.
FindFind2
1
1lim
1x
xx
SolutionSolution
Here is a general principle:Here is a general principle:
2
1 1
1 11lim lim
1 1x x
x xxx x
1
lim 1x
x 1 1 2
ReviewReview
Using the Limit Laws to calculate Using the Limit Laws to calculate limitslimits
Additional Properties of LimitsAdditional Properties of Limits Direct Substitution PropertyDirect Substitution Property
Lets look at :Lets look at :
•You must do some algebra You must do some algebra or trig. To simplify first:or trig. To simplify first:
What happens when you evaluate this next function?What happens when you evaluate this next function?
You get what is called an You get what is called an Indeterminate formIndeterminate form (it can not be determined) like (it can not be determined) like 00//00
•Now plug and chug!:Now plug and chug!:•The limit is 3The limit is 3
3
1
1
1limx
x
x
2
1
( 1)( 1)
1limx
x x x
x
2
1( 1)lim
xx x
Further Limit PropertiesFurther Limit Properties(if you think it would help you, then writ it down, if (if you think it would help you, then writ it down, if
not they are on p. 109)not they are on p. 109)
Applying the Product Law repeatedly Applying the Product Law repeatedly with with gg((xx) = ) = ff((xx) gives the following ) gives the following Power LawPower Law::
Here are two obvious but useful Here are two obvious but useful limits:limits: