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well no it doesn't rule is a tool that allows us to not documents quickly and
effectively
uh... let's talk about way into it to see
whether we actually still need to do that because we really haven't looked at
me and it's for a while
uh... we know that you're getting the message is rather cumbersome right i
mean we were very happy once we have a different issue rules
so we didn't have to use different washington work
uh... and the reason it's it's congressman scores the opportunites a
long time
out i think it we also mainly use them is to put the reviews on the son of
theoretical foundation
uh... instead i mean since then how often have we use good news that how
often and we used to read it
we can't just abandon limits because in the sense that you tell us about the
long-term behavior function
anti is something that is typically interesting and models you want to know
whether uh... the
object or is easy to make it is a stabilizing or not
uh... and some models we are also they still interested in what happens in
europe or notify by zero
so uh... units will stay with us there
not
uh... instinctual to calculate an estimate it s
indeed the grills and foundation for both but
uh... once we've got the resembles we work with us
op bak
they will stay with us in some capacity and so it'll be nice to the tool that
simplifies the
the competition
all key let's take a look at an example here's a quarter
off two quad products
in value no well and we we computed limits you got to do the express h
uh... in that age because you don't
you track look for factorization for cancellation
uh... that's magnify this
in what we see is
those two blood products
course they go through zero
at to otherwise tinnitus rather
easy to compute
and uh... what they see is that as we magnified these functions
because a differential functions
under sufficiently high magnification is
these functions start looking like streetlights
let's see if we can compute
the limit off the quotient off the tension box
so that his plan of the numerator and affects people's x squared minus six
explodes eight
well you take the derivatives that's two x months six
the derivative uh... to would be two times too which is four months six which
gives me too
and so
weekend in athens acts to would be to time six months to go to scrap the
pretzel
ends
unit
uh...
is zero
at tunes that should be
negative to their hang on just a minute
okay
now it's a native give
uh...
we find the tension kind of geo thanks
act
uh... x you texaco's experiments for
and it would be dead jeep radicals to expand the perimeter to his four
and the change plans four times excellence to wiggins zero act
too just like country
instructor itself
so the next
off of the quote riveted lines while you just cancel the excellence to use in the
magazine one-half
and if we compare that with the clean it off the actual quotient will you factor
the two functions and x great minus six explicitly in six months to
times
xmas for
ind exclude months for is exposed to expressed in the eight months to
schedule we get the limit of excellence for rates but still
and s
exposed to
that ends up being
native to for purchasing one-half and that's just saying right
now that's not an accident that name
from the picture we already saw that that shouldn't be an accident
and uh... will use it to you
it's multicultural
and uh... win effigy or differential functions in a surreal number or
infinity wall as infinity
uh... if the attempt to compute it into the courtroom after-effects of the
g_o_p_ exactly
beach an expression of the foreign zero the zero
or do you feel very fitting which are the first
two examples of
indeterminate forms that we're seeing it
i think the limit of the quotient hypocrites over to texas to say an
estimate
off the portion of the derivatives ex-prime fixed
overdue products
uh... invest provided that the limit on the right side exists in incident on the
right side doesn't exist then
minimax doesn't exist yet
this rule applies to one side unit still so if we can only approach from one site
that's not a problem
in the uh... the only priorities we must show we don't confuse this with the
court will rule
uh... typical mistake that happens i've seen happen on tests
is set at people use of the child when they should be a supportive role
or the use of force romantic when they should use all beat all all they're two
different things
so
proving what we can prove not the whole thing but these special case
oscillates consider the case that he's a number and as often equals g_o_p_ pulls
you up
so it's a quote a zero a zero
and the limits of attacks incorrect over g_o_p_ exceeds the limit office
excellency roper geo excellent serum
which is that ended up an excellent profitable thank you for talking
giovanni
it now we can use
the mean value theorem which says that type of excellence that okay yes
if prime of c_d_f_ for some c_f_ between x and eight times excellence and
int
do you have xmas g_o_p_ is g promotes e_g_ for some c g between x and a
times
excellent saying that makes mondays can easily get to the middle of the greatest
and as a discussed today
see a fancy g cody and to get
it's
x custody of prom affixed wifi g profits
okay
the enough that story is not proof
youth
uh... little
uh... unclean here because
it's c_f_ ncgs go at the wrong way and at significantly different rates for
city
these need not worry
there is a generalized mean value theorem that says we can't use the same
c
it's easy
uh... we're not going to go back deeply into theory socal just
be okay with this
quickens unfortunate cited iriarte
let's look at some examples and jj
well right now it'll be just was made for portions
find the limit as its goes to zero sign five x the conquest sign
activites bike two x
if you follow because we realize this is the rover zero detector additives he end
up with the limit of its goes to zero
five course and five eighths divided by two
and that's why i asked
does that really work
well if we're looking at the graphics here is a function
and the limit
five yes
now that was dialogue quakers and will begin at the beginning of the class and
in fact
determinants of
functions of portions that involved rate functions
i really hard you can do with pattern recognition after you know with the
democrats sign it's over x's
typically what you do is you use the veto so here we are we already have
conquered a problem
that would have been rather nasty before
if we're looking across that we had before and academic essex goes to a
expect us to excellence three-d_ graphics wants one
well let's see at at one that's zero zero so here's the limit
we find that in the health of the residence which of a group of two x of
xc request to excellence three strikes has to do that too
derivative off
x months one is one
so they know that as it goes to blankets for
uh... is that right well-kept here's the function years old and the whole house
is explore debt
compared to the right off the quotient off
toqueville acts well that's where things got really nasty and
are computations earlier well let's see what caused the latest really get
nine minus twelve plus three which is zero
int he read it
eighteen
minus six months twelve x zero also sort of the gospel can be applied zero zero
so we take the limit so
derivatives off expect us for expressed freest which transformed
the group of two x squared plus two weeks months twelve is four x trust him
if we can make it screaming down a little cricket zero the zero
week it
negative to divided by made ten right
hints that is one-fifth
inspects point to an acquittal cryptographic swell years function years
make three appears white
pointed
great works rather well done
let's see now those were were always i mean we computed limits right
beacause
that's what we had two expansion with aerobics plus peru date we have to
rationalize
numero denominator this case and now they're well let's see
at eight it's your was here over the top lies
so we take the limit
off derivatives for a girl excellency is one the derivative of rude xmas rate is
one of the two verdicts
estimate is exposed to a two hundred expat ends up being two times were eight
which he is four times were too
that's approximately five point six five
isn't
graphically
uh... eight
here's five
five point six something here
it works
if we can check now there is certainly couldn't handle before
and here's where i
without series and we don't have series yet
identity cancellation tracked the pledging cds
ethics code two zero
xc recruits to zero
key to the exit goes to one exposed as a result
one minus the reminds one is two zero zero
well they did they just
take the limit of the quotient
the computer industry bhobeck squared
two x
the drill each of the excellent six months months each six months one
logo i believe still can't compete with that because this is still zero ba zero
patch
multicultural
applies to that one too so we take the unity attitude read it again
during the gulf to exceeds two hundred of each of the xmas wannabes each of the
x
this image which we can we can compute now because there's ends up being to
over one
that still
and uh... ah graphically well here's the function and at fear of that same really
as a whole at height
what else can we do well let's take a look at that
uh... mother
unit off
quite products
odyssey actively get four minus two nd classics that zero numerator
intricate
four minus a plus for that's the road to nowhere so we can
apply to be cultural
into the derivative of excellent minus five extra sixty s
two weeks months five
letter into a expert months for expressed forties
to excellence for
when we get we get
any may have won a narrator if we get easy read the letter
so that it does not exist
and that means that the middle of the or regional quotient also does not exist
and graphically
if we're graphing the function we see we've got aboard alas it out
at that point which means they were included in that sense
in the car turns off
units as long as we compete with impunity that just means
that indicate to make sense
okay what else do we have
highly yay georg here is the limit is exposed to infinity
if we take that into six goes to infinity affixed to the seventh plus
five forex to the fifth months thirteen divided by three x to the seven th
my visits to the six plus
twelve x squared
well we expand with one of exit the seventh we end up with the limit being
the demographics of the seven over to the seven which is one in four x for the
fifth over to the second which is for the exquisite
niether t_-rex in seven right here
three x two seven or eight seven seven texas three may affixed to the six
erector set seventeen sesame one x twelve x squared
times when i read to the seventh gets us twelve over taxed to the fifth
and as a cs goes to infinity describe this kind of statements go against his
hero
so then it is
one-third we had to use the white house rule
we didn't in spot we shouldn't
deaconess
if we wanted to do this we would get infinity over infinity
for the first sixty religious retaking really have
new
patients to take all those derivatives
it turns out the certainty which may be
the old way to compute the nimitz
still is preferable
and of course schizophrenic result because well
used point five years point to five so one third of the boat there and that's
what the words on last dot
off this function
should be
let's take a look at another limited infinity
and on
let's just see
if we could use the beatles wrote here just just to see what could also
happened that's the most pathological thing that can happen but look it up
well it's going to deal with it so we can apply to pickups room
take the dribbles numerator which is three x squared take the director of the
calendar ok chain pool right went over to tell us what's inside
considering you know the inside which is six x two beth
use it to five six over
two x three c gets critics of the fifth year of the threes cancel
unix killed in the denominator and thus the moves into the numerator
so
we end up with
okay i guess at the three days of sex and get the two was alright but we end
up with
the limited exclusivity ultimately upward excellent six plus one tube at
the execute
which is exactly the receptor about this thing
well guess what if you look at how some of that
you end up
with the same thing that you start with so here's one where
milpitas rule isn't playing t_v_ use
it just doesn't work out
in the old-fashioned way of course you expand with one of ur execute
you get the limit of eight skewed or execute divided by route x two thirds of
excellence express
business card accidents whatever exit six right
and so that gives us an ethics coast infinity one over spirit one-plus
one of x six and that's one
and uh... if we craft this function
yep that works on glass of chilled
now let's look at any time unfortunately pf
done so far as we've looked at quotient i think i thought those many of those
quotient actually out of
the first minute presentation
that we
uh... worked our way through
is that shows that
uh... as long as were allowed to use the religious overtones for a week
replaces a lot of
tedious stuff
we've also seen that will be cultural
uh... allows us to compute somethings
that we couldn't compete with elementary methods
now let's take a look at intimate forms
even the computation which will be valuable terms when the plane in the
number
just needs to zero a zero toward finity over infinity
or zero times infinity
or infinity minus infinity
was never too busy roads or want to the infinity
infinity zero
that is calling indeterminate four
well he we've seen these two already
zero times infinity if the first time records may five two zero two and one
quarter infinity if the certain factor was really fast to infinity
because to go to phoenix with a into it
infinity was infinity well depending on how fast these go that could be in
number also
zero two one zero
algebraically
is typically defined to be one but depending on
how foss things go here
this exponent good still pull this
operates if the
base goes really fast zero
then this could be called towards zero
and if the exponent goes
uh... fast enough to zero well-connected it gets pull to one but it still isn't
imminent
one to infinity
this goes to one from above so it's like you because i wanted to try to really
big exponent correlate with impunity
infinity zeros
well if the pace was really quick to infinity even the racing to the rest of
the book back to one
there's all sorts of things
that could happen with these things these cannot be just
herself with some quick rule that says this is always the following
for example
we want to go get the limited six goes to zero one of our sign x minus whatever
x
well that's infinity
minus infinity
actually what i would like to have hearings i would like to have
right sided limit because otherwise we might have some trouble
with megan signs let me take a look at that quick
mousa writes i do not want to be careful here
with the negative signs actually turns out that author units from either side
are saying dot technically because of the trouble with nixon to what we have
to
computed separately
even though the computations are basically the same
right sized unit off whatever sign excellence whenever exiting the talent
form it's infinity
minus infinity
and i well what do we know this is not a quarter of a kid applied oakley cultural
but we can of course
take a difference of portions of turns it into a quarter by finding the common
denominator
which is fixed on scientific sense so then we get
xlr exciting explains
exciting x
excel
minus sites of excellence
insulates the bright side and an cyril x months identify by
eckstein sign it's now we have zero divide tear up
so we
take the derivative which is one might ask was alexander numerator int
denominator bs product will right through that makes this one
times side-effects plastics times group of science which is
was on its
and at zero we still end up with one minus one divided by its european zero
so
we use the beatles again
that there is a table one is zero three of the negative wilson is
function is to sign function
in here we get through the roof sign x's or sonics
hearings product will again
director to resolve the quiz signed exiting it's accident minus sign
eckstein sex
plastic or expense one
course i next now we have zero in the numerator
into we have one minus zero plus one and in a minute or so thats
zero to fight back to the stores around the numerator
just consistent zero
int
uh... the
graphics actually
bear that out
if you can see from the true site unit
zero here but we were technically
because we're coming in from the negative side we would have had megan
profanity minus native infinity
as an indeterminate form it would have
technically have to do a separate computation
okay what else can we give
we can look at
things like extends into the matrix right the alls
uh... rule of thumb that exponential growth or exponential decay
beats polynomial growth that's something we can
justify no
so this use
an indeterminate form off the form infinity times zero
well take the limit
turned it into a portrait now it's infinity over into a deal because
replies
the the middle of the derivatives it's an elf
the periodontics is wondering who agreed to be a to z to the exxon cid image
off one of each of the x
which is the limit is exposed if any of you could get x which is two zero
intake in graphically that's what happens the x
tries to push this out but as soon as the exponential decay really gets going
this function to casework zero very very rapidly
if this is something that you can do
for people who will be chris after you take enough to reduce off those
polynomials
you end up with something defied the exponential
ex-cop will notify exponential will always go to see right through me
now let's take a look at something else to take the right here june the twelfth
is where it takes time so our topics
okay wesat that's infinity times minus infinity while the minus
when it doesn't matter that much because packwood out
in so
what do we do
well the need to turn that into a quotient
into their are
two possibilities to turn that into a courtroom weekend
make that for the middle fruitcakes divide it by one all for
that already mosaics and we have really ugly derivatives
whether they want to do this when i get rid of all over them
so it's probably better to write these estimates
of logarithm
divide by one overrode xlr experience one-half because we know
this one become central to differentiate this was one of her ex veteran of this
was one of x
whereas this one is just another power interpreted
okay so take the derivatives which is about because we end up with
minus infinity divide it by committee now if we take the the nazi era
we take the great if we get one of rick's divide it by power rule here
native-born half-time six to three hats
that that at first we want simplify it
into well because we've got a native exposed here
we end up with low-key one over net if what happens next to him
exterminator three has a six to three has won over xmas nixon na letter
now we can canceled its ends up being
well okay
the cancellation i forgot to tell you
this is the limit is exposed to zero negative two weeks for the one hand
right
and then that ends up being zero
ansatz
indeed but the function does look at this
logarithm
then he tries to pull the functionally
intern at the last
at the last hour
square root
calls at back to the origin
uh...
what else can we do well we can look at zero two zero
that's not a quotient dot that's not something that we can
tune into a quotient once we
uh... or
by october
so the first thing we do to computing the florist
okay clobber them next to the verdicts
that's rude x
times all over the next it was just computer that document
as exposed as they were from the right one zero
in so that means that the limit of x to the root x ethics courses or from the
right
is the limit off he can exclude ex-wife that says test
realizing
that
uh...
excimer excuse any to the
inexorably expression of interest
plugin
even though that that limit
use or
we know that the exponential functions continuously people than you can see
exponent
in reno that that limiting the export of his ears to get you to the zero switches
one
that turns out to be a case began tries to put away put them at the last minute
in act equals
towards what
okay
uh...
in we can also do that for a minute diesel
let's take a look back any limit where we have what we have needs hero to
infinity
uh... well makes goes to infinity
one other excuse to zero
in small numbers raced in large numbers become even
cid image zero
and that's because you're right uh... zero infinity is not the intended for
right
that was actually
on this
uh...
and so
uh...
i hope it was not on the list otherwise i'd have to correct the slides and i
would oppose correct size
but yes it was going to be
when something gets small when something is
smaller than wanting to raise it to a large number that's going to go to zero
and if
the number itself goes to zero to just goes to zero even faster so
be careful
sometimes looks like the instrument falling kit
you don't need to apply loopytoe
okay graphically check and i think just balance is two zero netscape
okay now let's also work with parameters i decided on these later slides i'm
going to
take the sermon on what we need to
weapons parameters out to me by now
being in college for a little bit these new crawling seeing your other classes
that you have to deal with parameters
let's take a look at this one that is exposed to infinity one clasping over
x_-rays to the x
so this is now
one between fifty and dot is an indeterminate form
in cell again because it's a power we first take them over them so
the news exclusive video from all over them off this thing well logarithm pulls
the power outlet
sorry for the unclean
marking here
but basically that's the news exclusive his x
times event
one plus annual ranks
if you need to turn that into a portion a we do that by taking the time it is
exquisite infinity
every one of our one-plus
ever excellent by one of x
if you did x over one over the event
would get so many of the expressions and realized melanie is trying to track the
other way
uh... but here now we take derivatives okay that revealed a lower than this one
over waivers in sight
which is one of one class
eighty overextending
times and rebuild the insides of credible one is the road ridable
anorexic
erect square brutal one x is made of one of the x squared
you can see here the native-born over x
squier actually cancelled
and so this is the net tax goes to infinity pool
eighty which is quick remains in the numerator
uh... any over one plus
eight overexposed pics coast infinity that ghost eight
in so that means that the limit of one-plus ever explains to the x
incident of each of the airline
of what we have previously
into report limit
into the exponent that speed to create
that is actually
rather nice not is any standard formula for the exponential function
which we can now prove
fairly quickly
this is not how the exponential function
is represented numerically
inside a computer
it is sometimes
basically when we never these kinds of
units come up we
know now that yet we defaulted to the exponential
graphically speaking
i dash that's it
uh... which any and using here
uh... or rich
electrical equipment to infinity
explained to infinity so the input actually is a hearing on x
dot uh... basically
bs
i that v
exponent go to infinity davies functions kit
closer and closer
exponential functions
that's the beatles rule
basically what it does for us you speaks it shows as a couple of computational
tricks
probably aren't too bad to have because it keeps us working with
morality bryant remember indicated that with the u_n_ freakin
have to computer limits
if you're in engineering
on trying to rebuild
and if you're in the science of specifically in physics only
odds are only
slightly
higher that you do
but nonetheless
uh... he
techniques that we have here the power of recognition
theories ways in which we can
turned functions around and going to turn into something
that we can't digest with theoretical methods
that's what ought to stick with you as well as of course as you doing
homeworks and as you're passing tests involved this stuff manipulations
four loopytoe through
itself intent of course
if is rather beautiful to see how
the u_s_ oftentimes impact zones that we had
helps us to compute units
that previously
could not come shoot directly
so we thought you could knock out plenty of notice on your homework