opponents of taylor and l'hospital ?
amy666
i was then told by certain people that they dont like l'hospitals rule nor taylor series.
they said l'hospitals rule was overrated !
can those people plz explain what is it that makes them opponents of l'hospital rule and taylor series ??
i dont see a downside of l'hospitals rule ?
and i was quite amazed to read that on a mathforum from even relative regular posters.
regards
tommy1729
tamiry
> some while ago when talking about limits , i used l'hospitals rule or taylor series.
>
> i was then told by certain people that they dont like l'hospitals rule nor taylor series.
>
> they said l'hospitals rule was overrated !
>
perhaps people don't like over-using it when some effort could give
you the result.
for instance:
(1 - cos(x)) / x^2
you could either use the rules or simply do the trigonometry.
> can those people plz explain what is it that makes them opponents of l'hospital rule and taylor series ??
>
> i dont see a downside of l'hospitals rule ?
>
> and i was quite amazed to read that on a mathforum from even relative regular posters.
>
refer us to the source, than.
> regards
> tommy1729
Denis Feldmann
> some while ago when talking about limits , i used l'hospitals rule or taylor series.
>
> i was then told by certain people that they dont like l'hospitals rule nor taylor series.
>
> they said l'hospitals rule was overrated !
>
> can those people plz explain what is it that makes them opponents of l'hospital rule and taylor series ??
If you were not such a crank, you could get explanations. As it is,
ponder on lim (x\to infinity) ln(x^2+sin (x^3))/sqrt(x+cos x). Is really
L'Hospital or Taylor such a good idea ?
amy666
> amy666 a écrit :
> > some while ago when talking about limits , i used
> l'hospitals rule or taylor series.
> >
> > i was then told by certain people that they dont
> like l'hospitals rule nor taylor series.
> >
> > they said l'hospitals rule was overrated !
> >
> > can those people plz explain what is it that makes
> them opponents of l'hospital rule and taylor series
> ??
>
> If you were not such a crank, you could get
> explanations. As it is,
> ponder on lim (x\to infinity) ln(x^2+sin
> (x^3))/sqrt(x+cos x). Is really
> L'Hospital or Taylor such a good idea ?
since x goes to oo we can disregard the sine and cosine.
then we have 2*ln(x)/sqrt(x).
and both l'Hospital and Taylor can be applied here.
>
> >
> > i dont see a downside of l'hospitals rule ?
> >
> > and i was quite amazed to read that on a mathforum
> from even relative regular posters.
> >
> > regards
> > tommy1729
regards
tommy1729
Gerry Myerson
<7402283.12099151708...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
> some while ago when talking about limits , i used l'hospitals rule or taylor
> series.
>
> i was then told by certain people that they dont like l'hospitals rule nor
> taylor series.
>
> they said l'hospitals rule was overrated !
>
> can those people plz explain what is it that makes them opponents of
> l'hospital rule and taylor series ??
>
> i dont see a downside of l'hospitals rule ?
There's an objection to using l'Hopital to do
limit as x goes to zero of (sin x) over x.
The objection is that to use l'Hopital, you need to know
the derivative of sin x --- but to know the derivative
of sin x, you have to know the limit as x goes to zero
of (sin x) over x, so using l'H is circular (no pun intended)
reasoning.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
[Mr.] Lynn Kurtz
<ge...@maths.mq.edi.ai.i2u4email> wrote:
>There's an objection to using l'Hopital to do
>limit as x goes to zero of (sin x) over x.
>
>The objection is that to use l'Hopital, you need to know
>the derivative of sin x --- but to know the derivative
>of sin x, you have to know the limit as x goes to zero
>of (sin x) over x, so using l'H is circular (no pun intended)
>reasoning.
I don't recall ever seeing the limit for sin(x)/x developed in any
calculus book by using L'Hospital's rule. Have you?
--Lynn
Gerry Myerson
No, not in any book - but on student papers, yes, many times.
The objection is really to assigning that limit as a problem
when you're teaching l'H.
lwa...@lausd.net
I think that tommy1729 raises a very interesting point here.
I know that I've seen some of these posts that actually recommend
that L'Hopital's rule not be taught in calculus courses. I kept trying
to find such posts using a Google search. I couldn't find them, but
I know I've seen them. (It may have been Dr. Herman Rubin who
discourages L'Hopital's rule, since he's definitely outspoken on
what should and shouldn't be taught in school, but since I can't
find the thread I can't be sure whether he's the one.)
I actually agree with tommy1729 here. I believe that we should
continue to teach L'Hopital's rule in calculus classes.
In this thread below, Dr. Gerry Myerson does give a valid reason
why L'Hopital's rule in certain situations is flawed:
Myerson:
> There's an objection to using l'Hopital to do
> limit as x goes to zero of (sin x) over x.
> The objection is that to use l'Hopital, you need to know
> the derivative of sin x --- but to know the derivative
> of sin x, you have to know the limit as x goes to zero
> of (sin x) over x, so using l'H is circular (no pun intended)
> reasoning.
But I believe that in classes that are taken mainly by
students whose major is not mathematics, such as
freshman calculus, complete mathematical rigor is not
always necessary. The goal of the class for non-math
majors is _computation_, not _proof_. Save the proofs for
upper division classes taken only by math majors, not
engineers or physics majors.
Suppose a high school senior taking the AP Calculus exam
this month sees the following problem:
lim (x->0) (sin x)/x =
A) 0
B) 1
C) -1
D) does not exist
If I were this student, I'd simply use L'Hopital's rule and find
the anwer to be (cos x)/1 = cos 0 = 1 in seconds. Why
should I care that the limit is used in the proof that
d/dx (sin x) = cos x? The question didn't ask to prove that
d/dx (sin x) = cos x -- it asked to find the limit, and the
fastest way to find the limit is L'Hopital's rule.
I agree that there are cases, such as lim (x->0) P(x)/Q(x)
where P and Q are both polynomials, where there are
other methods (such as long division) that are more
appropiate than L'Hopital's rule. But whenever I see
lim (x->0) P(x)/Q(x) where P or Q is any function but a
polynomial, then I'm using L'Hopital (provided that it can
actually be used) -- despite the fact that limits such as
lim (x->0) (sin x)/x or lim (x->0) (e^x-1)/x are needed to
prove the differentiation rules for sin x and e^x.
In fact, to this day I don't remember how to prove the
differentiation rules for sin x and e^x -- yet I still recall
lim (x->0) (six x)/x = lim (x->0) (e^x-1)/x = 1, thanks to
L'Hopital's rule. Except for polynomial functions, I
cannot differentiate any function from first principles only.
I still recall my multivariable calculus professor, who
referred to L'Hopital's rule as the one rule that every
student from calculus likes. (He jokingly called the Mean
Value Theorem "the much hated Mean Value Theorem.")
On the other hand, there are some teachers (thankfully
not any teacher I had, but someone who posts here at
sci.math) who would emphasize using the definition of
the derivative and the Riemann integral to differentiate or
integrate any function, and not teach the differentiation
rules for any function, not even polynomials. This is
definitely inappropriate for high school, or for any
class populated by non-math majors. Using the definition
of the derivative to differentiate a polynomial is an
error-prone process. Using the rules is much less likely
to result in mistakes.
porky_...@my-deja.com
> In this thread below, Dr. Gerry Myerson does give a valid reason
>
> this month sees the following problem:
>
> lim (x->0) (sin x)/x =
> A) 0
> B) 1
> C) -1
> D) does not exist
>
> If I were this student, I'd simply use L'Hopital's rule and find
> the anwer to be (cos x)/1 = cos 0 = 1 in seconds. Why
> should I care that the limit is used in the proof that
> d/dx (sin x) = cos x? The question didn't ask to prove that
> d/dx (sin x) = cos x -- it asked to find the limit, and the
> fastest way to find the limit is L'Hopital's rule.
>
There are other ways to show that d/dx(sin x) = cos x, without
computing the limit of (sin x)/x first. If you start with defining
complex exponentials e^z as the series \sum_{n=0}^{|infty} z^n/n!, and
then define e^{ix} = C(x) + iS(x), it immediately follows that d/dx(S)
= C, by observation of series representations of C(x) and S(x). The
only remaining part is to show that both C and S match the regular cos
and sin we learned in high school. Once we have this in place, we can
certainly use the L'Hospital rule to show that the limit is 1.
BUT practically all Calculus textbooks derive the limit of sin(x)/x
first, using pure geometric means, and *then* use it to show that d/
dx(sin) = cos x. With this in mind, using L'Hospital rule to argue
that the limit is 1 *is* a circular reasoning. That's why you should
care how d/dx (sin x) = cos x was obtained in the first place. (Of
course there is a reason why Calculus does not do it 'series' way: you
have to have all the series related staff in place, so that's normally
done in analysis. Baby Rudin does it this way.)
> In fact, to this day I don't remember how to prove the
> differentiation rules for sin x and e^x -- yet I still recall
> lim (x->0) (six x)/x = lim (x->0) (e^x-1)/x = 1, thanks to
> L'Hopital's rule. Except for polynomial functions, I
> cannot differentiate any function from first principles only.
>
Well, there are different ways to define e^x. Sometime you start with
ln x first, as the integral of 1/x, then define e^x as inverse
function of ln x, then all the properties including d/dx e^x = e^x
follow. Of course in case of series representation, the fact that d/dz
e^z = e^z immediately follows, by observation.
Dave L. Renfro
> Suppose a high school senior taking the AP Calculus
> exam this month sees the following problem:
>
> lim (x->0) (sin x)/x =
> A) 0
> B) 1
> C) -1
> D) does not exist
>
> If I were this student, I'd simply use L'Hopital's
> rule and find the anwer to be (cos x)/1 = cos 0 = 1
> in seconds. Why should I care that the limit is
> used in the proof that d/dx (sin x) = cos x?
> The question didn't ask to prove that
> d/dx (sin x) = cos x -- it asked to find the
> limit, and the fastest way to find the limit
> is L'Hopital's rule.
I agree with you, although for this particular limit
students shouldn't have to rely on any rule. This
should be one of the "common limits" they know by
heart (like the multiplication table), because it
is so useful in finding other limits by algebraic
manipulations, something they would have had a bit
of practice with in the first few weeks of their
calculus class.
I think the main reason some people don't like the
use of L'Hopital's rule is not this issue, but rather
because it is essentially a "black box" process
that typically sheds very little additional insight
(for example, inequalities used to set up the squeeze
theorem allow for errors and/or convergence rates to
be estimated) and tends not to reinforce any other
concepts except for the mechanical process of using
short-cut derivative formulas (in fact, this is what
I primarily used L'Hopital's rule for -- a way to get
students to review derivative short-cut formulas).
Dave L. Renfro
amy666
> "[Mr.] Lynn Kurtz" <ku...@asu.edu.invalid> wrote:
>
> > On Mon, 05 May 2008 00:27:58 GMT, Gerry Myerson
> > <ge...@maths.mq.edi.ai.i2u4email> wrote:
> >
> >
> > >There's an objection to using l'Hopital to do
> > >limit as x goes to zero of (sin x) over x.
> > >
> > >The objection is that to use l'Hopital, you need
> to know
> > >the derivative of sin x --- but to know the
> derivative
> > >of sin x, you have to know the limit as x goes to
> zero
why do you need the limit as x goes to zero to compute the derivatative of sin x ??
its not that hard ?
1) its very well known
2) you can use taylor series
3) you can use nilpotent numbers h =/= 0 and h ^ 2 = 0
> > >of (sin x) over x, so using l'H is circular (no
> pun intended)
> > >reasoning.
> >
> > I don't recall ever seeing the limit for sin(x)/x
> developed in any
> > calculus book by using L'Hospital's rule. Have you?
well actually i have.
dont remember which one , and it might not have been a good one , but im sure i have.
>
> No, not in any book - but on student papers, yes,
> many times.
> The objection is really to assigning that limit as a
> problem
> when you're teaching l'H.
why ? you can use that example to demonstrate l'H.
>
> --
> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for
> email)
regards
tommy1729
The World Wide Wade
<13974fd3-8184-4deb...@u6g2000prc.googlegroups.com>,
lwa...@lausd.net wrote:
No, the main goal is not computation (you can do these things on a
computer), and it's not rigorous proof. It's understanding and
intuition.
> Save the proofs for
> upper division classes taken only by math majors, not
> engineers or physics majors.
Fine, but that has little to do with the dislike of many here for the
way LHR is invoked.
> Suppose a high school senior taking the AP Calculus exam
> this month sees the following problem:
>
> lim (x->0) (sin x)/x =
> A) 0
> B) 1
> C) -1
> D) does not exist
>
> If I were this student, I'd simply use L'Hopital's rule and find
> the anwer to be (cos x)/1 = cos 0 = 1 in seconds. Why
> should I care that the limit is used in the proof that
> d/dx (sin x) = cos x? The question didn't ask to prove that
> d/dx (sin x) = cos x -- it asked to find the limit, and the
> fastest way to find the limit is L'Hopital's rule.
No, the fastest way is simply to know this result. It is one of the
fundamental bedrock limits of calculus. If you don't know that limit
like the back of your hand after first semester calc, something has
gone wrong.
> I agree that there are cases, such as lim (x->0) P(x)/Q(x)
> where P and Q are both polynomials, where there are
> other methods (such as long division) that are more
> appropiate than L'Hopital's rule. But whenever I see
> lim (x->0) P(x)/Q(x) where P or Q is any function but a
> polynomial, then I'm using L'Hopital (provided that it can
> actually be used) -- despite the fact that limits such as
> lim (x->0) (sin x)/x or lim (x->0) (e^x-1)/x are needed to
> prove the differentiation rules for sin x and e^x.
>
> In fact, to this day I don't remember how to prove the
> differentiation rules for sin x and e^x -- yet I still recall
> lim (x->0) (six x)/x = lim (x->0) (e^x-1)/x = 1, thanks to
> L'Hopital's rule.
Forget about proofs, that's a red herring. The limits here are nothing
but derivatives themselves! lim (x->0) (e^x-1)/x is by definition the
derivative of e^x at 0. That's it. If a calc student doesn't know
that, then he has learned very little.
> cannot differentiate any function from first principles only.
>
> I still recall my multivariable calculus professor, who
> referred to L'Hopital's rule as the one rule that every
> student from calculus likes. (He jokingly called the Mean
> Value Theorem "the much hated Mean Value Theorem.")
>
> On the other hand, there are some teachers (thankfully
> not any teacher I had, but someone who posts here at
> sci.math) who would emphasize using the definition of
> the derivative and the Riemann integral to differentiate or
> integrate any function, and not teach the differentiation
> rules for any function, not even polynomials. This is
> definitely inappropriate for high school, or for any
> class populated by non-math majors. Using the definition
> of the derivative to differentiate a polynomial is an
> error-prone process. Using the rules is much less likely
> to result in mistakes.
But the rules for differentiating polynomials are something you can
teach an 8 year old. Such rules are trivial; any dull mind can learn
them. Again, your citation of some mad teacher's practices have little
to do with the main issue.
Rob Johnson
amy666 <tomm...@hotmail.com> wrote:
>> In article <4VseSEovwEPDM9...@4ax.com>,
>> "[Mr.] Lynn Kurtz" <ku...@asu.edu.invalid> wrote:
>>
>> > On Mon, 05 May 2008 00:27:58 GMT, Gerry Myerson
>> > <ge...@maths.mq.edi.ai.i2u4email> wrote:
>> >
>> >
>> > >There's an objection to using l'Hopital to do
>> > >limit as x goes to zero of (sin x) over x.
>> > >
>> > >The objection is that to use l'Hopital, you need
>> to know
>> > >the derivative of sin x --- but to know the
>> derivative
>> > >of sin x, you have to know the limit as x goes to
>> zero
>
>why do you need the limit as x goes to zero to compute the derivatative of sin x ??
>
>its not that hard ?
>
>1) its very well known
Proof by common knowledge. Right below proof by intimidation.
>2) you can use taylor series
How does one develop the taylor series for sine without knowing how
to differentiate sine?
>3) you can use nilpotent numbers h =/= 0 and h ^ 2 = 0
You lost me there. sine is defined on the real line and can be
extended to the complex plane, but neither of these fields have
zero divisors.
>> > >of (sin x) over x, so using l'H is circular (no
>> pun intended)
>> > >reasoning.
>> >
>> > I don't recall ever seeing the limit for sin(x)/x
>> developed in any
>> > calculus book by using L'Hospital's rule. Have you?
>
>well actually i have.
>
>dont remember which one , and it might not have been a good one , but im sure i have.
It would be nice to see a reference. They are much more convincing.
>>
>> No, not in any book - but on student papers, yes,
>> many times.
>> The objection is really to assigning that limit as a
>> problem
>> when you're teaching l'H.
>
>why ? you can use that example to demonstrate l'H.
Probably because of the circular nature of the reasoning.
I have nothing against using Taylor series or L'Hospital's rule,
except when used circularly in a proof. If one has already derived
the derivative of sine, then there is nothing wrong with using that
result to rederive the limit of sin(x)/x. However, since this limit
was already derived, why rederive it?
I believe that the thread that started this was "Real Analysis Help
.. Sequences" when William Elliot said, "lim(x->0) (sin x)/x = 1.
That is calculus." and I replied, "That is circular, at least if
you are thinking of using L'Hospital." I would have kept quiet if
he had said, "That is pre-calculus." Pre-calculus generally covers
trigonometry and limits; definitely enough to handle this limit.
Pre-calculus also precludes the use of L'Hospital.
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
porky_...@my-deja.com
> >2) you can use taylor series
>
> How does one develop the taylor series for sine without knowing how
> to differentiate sine?
>
See Rudin, Principles of analysis, Ch 8, or see my posting above for
outline. Both sin and cos are initially *defined * as power series.
amy666
( forgive my shorthand notation )
> In article
> <13974fd3-8184-4deb...@u6g2000prc.googl
understanding , intuition and computation often go hand in hand.
sure , but users of l'hospital know this result from the back of their hand too.
they use l'hospital to show it to others , not because they dont know it themselves.
sure , but thats exactly one of my points.
let me explain : when they say ( and they did in this thread ) you need the lim of sin(x) to compute sin(x)/x ( at 0 ) , that limit is just the derivative.
just as you said lim = derivative.
so in fact it does not give a circular or counterexample of l'hospital, since instead of LHR -> needs lim , we get , in a way , LHR = derivative = lim.
and btw we can also apply taylor or h =/= 0 , h^2 = 0 to solve in a more algebraic way.
nostalgy i assume.
my prof often said : "everything (in the real world) is a wave"
to which i replied : " if you have a hammer , everything looks like a nail "
the matter is still unsettled , in fact it got more complicated over the years e.g. wave vs particle vs curvature in physics , discreet event vs continue wave vs attractors in certain economic models etc
regards
tommy1729
Bill Dubuque
There is in fact an _algorithm_ for evaluating limits of a wide class
of elementary functions that I discovered around 1980. In a rough
sense the algorithm works by computing certain generalized power
series expansions (or initial terms thereof), so it may be viewed
as an amalgam of applying taylor series and L'Hospital's rule. More
precisely it works by computing higher-rank valuations in certain
(Hardy-Rosenlicht) differential fields. There are probably only
a handful of mathematicians familiar with such techniques, so one
can safely say that almost all responses you receive here will
not be well-informed. See my prior post [1] and its links for
some nontrivial examples and further remarks.
--Bill Dubuque
[1] http://google.com/groups?selm=y8zk7js12ba.fsf%40nestle.ai.mit.edu