Re: opponents of taylor and l'hospital ?
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: Mon, 05 May 2008 12:29:27 EDT
lwalke3@xxxxxxxxx wrote (in part):
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
.
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