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Agenda: Feb.21• HW Questions• Droodle - what's the picture?.• Limits Review• 7.8 L'Hopital's
Evaluating Limits: Try direct substitution first.
These are ok: (for C"* 0)
C :) 000
Co -~ 00
00 + C -) {/O
00' C =
1
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Indeterminate Forms:0 00 00 00
0' , o ' , 00 - 0000 00
000, 100
, 0° 00·0 00°r
If an Indeterminate form results, :• If Rational: Factor/reduce• If Radicals: Rationalize the expression• If Trig functions: Apply trig properties to simplify
• L'Hopital's Rule• Use with ~ or 00o 00
First, let's review earlier strategies from Chapter 1
.1-o
Io
0-' - Io
2
o
..
[PCID©~@[fo . X3 -X XexL_1) )(..(Y-+l')(rl)
hill - .•..'"---o x~l x-I 1--- I - C)t. 7fJ
J.A-- ;(.(~ I) = I (I-+l) =-0"''':::;I ~
f4tO -2 0- -o - 0
RationalizelimlJ4 + x - 2 z: e 4-+X -2)( r,;;:; •.~) (~)( - 4)HO\ X . '1-- (r'1 " "2.\ -.:: ( f.iti .
,,)I( - ),_ I I I-7 vA ) - }..")o - =- - e: ~oa/' ''1.)t -(. (n:r::x -}) fiI+{) -2-
Three Special Trigonometric Limits
1) lim smx = 1x~o x
2) lim tanx = 1x~o x
3) lim 1- cos x = 0x~o X
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Limits at infinity
) I, 3-5~ - 1Ja un tV -.::...'X~Cf) 3~-1
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--k- r;b)
£~ _3--a.
- -~ ~-\(i: 2- j.,
--z,.. \'-:-z.}.. k
-]>( ~-"'- -
by the sandwichtheorem:
-/,
I, sm x 0nTI =X~Cf) X
-
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L'Hopital's is a special theoremthat can be applied to the
indeterminate forms:t
oo
coor
co
Can often rewrite other indeterminate forms to looklike:
oo
coor
co
L'Hopital's Rule Also known asL'Hospital's Rule
If [ and 9 are differentiable and continuous and g' ex) * 0 onan interval, except possibly at c. then
If. [ex) 0 00
hm()=O or -x ....•cg x 00
We can determine the limit as follows:
lim [(x) = lim f' (x)x~c g(x) x-sc g'(x)
,
X
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Apply L'Hopital's Rule to the following: F\rs;- sho \A) ,'t-lJo~ for Y'1l J2- or 2!Q
lim (~ . Sin(7X)) = 'S11'1 (O) _ 0 " L ~X~O X '0 -- D ~O c o-rv ~r 'I
lim ( In x ) -:= ~ ( , \, 2!l- c Dx~l X - 1 l- I 0
~ / trq) ~ L.:-- / '/)(. J -=n I L-A-l 7 X-II tI) =[DL'Hopital's can be applied multiple times, as long asthe you sill have: ; or :
6
••.-
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Careful using L'Hopital's, can only be applied when theI" it i 0 00Iml IS: - or -o 00
(sin x )
lim =-X-71[- 1- cos x
B~ --rhUJ .AL> ~rJ1SJ! !~ 5 (I' X S''l (rr-)~1t- -
I-CD..sy-- {-CaS7I-
Indeterminate products: 0 . 00
lim x sin ~ - (;X:? I $ I,"" ~A ~ I?O #
x-+oo X -v-
A product can be rewritten as a quotient using thefollowing:
Example:
\S,,, 00 s-o :U----
100-t'
..
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Indeterminate Differences: 00 - 00
Get common denominator and try to convert to :
Example:
lim (_1_ _ 1 ) .::x~l lnx x - 1
~i,~I.
\ ~-I- ,,- -~'I.- -r-- - l
A .1 -I -/./'l-" I
o
o 00
o or 00
J.. - ~.e.n I I-I
L'I-t d-et5
I
z: l .u .- - .--. - ~-"Yo b
~Dt «<1D~,
~ ~- I-~~(X-')~)<.
LH';) - P_'. ,-J... I \7 ~;r x. ~ --
kx-t, -~ . ~I+l-~ o
\- j(-I~---~l J;ny.-t l - ~
8
..
L
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Indeterminate exponential: 100,000,00°,00 I~ -PC'll \
b = e1n b ~ {c,><.) =c
Iflirn lnf(x) = LX--7a L )
-fl ()I') ,ywf 5
roh ~ ~ F(x,l~() hrd Ad't (
When the variable is both in the base and the exponent,Be ca refu I!!!!!!
Example: 100 I
1~~(1+ ~r~ Q + ~) <>- ~ \ 00
LeA {:'(t<-):::: (I -+ ~ ') )(
L <C(x-) ~ »: ~ ( 1;- ~ )
1-1-)(-'
~(I-+~) .--0n ( I ) 0 (JOw Af ~L'H- .- ~-...L 0M
.s-> \ 1 =QJ--- - --, - \-:::: y..-) 00 14~ f ~ 0;, ,
~~ f()()=~1
'1--701
I'"
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Example: 00 Io
:::-0 :c~1. ----) ~'F(x) ~ x J-.n{ '1.1
[XJ:: -
00
~('f-) ~ X-IX .l.h"f.--) ~ {'(X-) - »:
~r-- -C x ~ 00o I::. ~
'I-I- ~ Vt.A.-e.Lx.->0a - -
( v: .00 ()t::J
~{Ill.
X~\- \ , ~@- -
f..~flo ~-::::;~ X 00
Example: 000 Ilirn xlix
x~oo
6
2
..
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J~w+(cos G - x )f =c ( Co s(~ -0) )0 ~ 0 D CT'•••
~1r f{x) ~(c oS L ~ -x)) xs-: c{ Xl ~ x, ~ ( (()S'(%-'L) l
x. ;0+ 1-.Lr{ C1> s('~ X)J - o·~ (c 0 sl:s!i.. ') 'C 0'"""
--SIf"\C!1/Z-Y
C os fT/~-x.
1
.-
Chapter 3
Lab on Indeterminate FormsIn this section we will be evaluating limits that may give results that may surprise you. For the followinglimit problems, make a guess as to what you think the limit will be and then evaluate the given expressionfor the values given and see if you think your guess was correct.
. (e3X -.1) I, I 01. lim -- =? -:. -'0 Guess: ----x-e O x 0To see if your guess is correct, evaluate the following:e':-t", 5, 'tqq
e':-t.,m 3, DLI S
e':-t..ooi - 3 I OD '1Now what do you think the original limit is? __ 3__
2. u (I I) _? -'- _.Jx~ lnx - x-I -. 0 0To see if your guessis fOr ect, evaluate the following:
Guess: _
L~x- x~1)1=1.0L
L~x- x~1)lx=1.00L
Now what do you think the original limit is? I S'3. lim(l +~)X = ? Guess: _
x~~ xTo see if your guess is correct, evaluate the following:
(l+~r=LOO
( 2)X1+-X x=LOOO
(l+~rx=lO,OOO 7.3 <Q t3
Now what do you think the original limit is? _
7, ~ '-I 5
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