warm-up 1-3: evaluating limits analytically ©2002 roy l. gover () objectives: find limits when...
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Warm-Up
1-3: Evaluating Limits Analytically
©2002 Roy L. Gover (www.mrgover.com)
Objectives:•Find limits when substitution doesn’t work•Learn about the Squeeze Theorem
Example
Find the limit if it exists:3
1
1lim
1x
x
x
Try substitution
Substitution doesn’t work…does this mean the limit doesn’t exist?
Try the factor and cancellation technique
Important Idea3 21 ( 1)( 1)
1 1
x x x x
x x
2 1x x and
are the same except at x=-1
Important Idea
The functions have the same limit as x-1
Procedure1.Try substitution2. Factor and cancel if
substitution doesn’t work
3.Try substitution againThe factor & cancellation technique
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
5Isn
’t th
at
easy?
Did you think ca
lculus
was going to
be
difficu
lt?
Try ThisFind the limit if it exists:
22
2lim
4x
x
x
1
4
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
The limit doesn’t existConfirm by graphing
Important Idea
If substitution results in an a/0 fraction
where a0, the limit doesn’t exist.
DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.
Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as the factor and cancel technique.
Try ThisFind the limit if it exists:
2
1
2 3lim
1x
x x
x
-5
ExampleFind the limit if it exists:
0
1 1limx
x
x
Try substitutionWith substitution, you get an indeterminate form
Try factor & cancelFactor & cancel doesn’t workHorrib
le
Occurrence!!!
The rationalization technique to the rescue…
Rationalizing the numerator allows you to factor & cancel and then substitute
BC warm-UpFind the limit if it exists:
0
1 1limx
x
x
Try ThisFind the limit if it exists:
1
2 2
0
2 2limx
x
x
The Squeeze Theorem
Let f(x) be between g(x) & h(x) in an interval containing c. Iflim ( ) lim ( )
x c x cg x h x L
lim ( )x cf x L
then:
f(x) is “squeezed” to L
ExampleFind the limit if it exists:
0
sinlim
Where is in radians and in the interval,2 2
ExampleFind the limit if it exists:
0
sinlim
Substitution gives the indeterminate form…Factor and cancel or rationalization doesn’t work…
Maybe…the squeeze theorem…
Example
g()=1
h()=cos
sin( )f
Example
0lim1 1
0
lim cos 1
&
therefore…
0
sinlim 1
Two Special Trig Limits
0
sinlim 1
0
1 coslim 0
Memoriz
e
Example
Find the limit if it exists:
0
tanlimx
x
x
0 0
sin 1lim lim 1 1 1
cosx x
x
x x
Example
Find the limit if it exists:
0
sin(5 )limx
x
x
0 0
sin(5 ) sin(5 )lim 5 5 lim 5 1 5
5 5x x
x x
x x
Try This
Find the limit if it exists:
0
3 3coslimx
x
x
0
Lesson Close
Write, in outline form, the procedures for finding limits when substitution doesn’t work.
Assignment
68/45 – 61 odd