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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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3

.-

Limits at infinity

) I, 3-5~ - 1Ja un tV -.::...'X~Cf) 3~-1

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4

--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

5

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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

2/22/2013

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'"

,v2/22/2013

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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