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Topic: L'Hospital is NOT Blind
Replies: 3   Last Post: Dec 1, 2004 12:33 AM

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J. J. Sroka

Posts: 85
Registered: 12/6/04
L'Hospital is NOT Blind
Posted: Oct 22, 2004 4:36 PM
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In a previous post, Karplus and Chapman both referred to "blindly"
applying L'Hospital's Rule.

Let me make a point about using L'Hospital's Rule by using a much more
poignant (thank you, Tommy Smothers) example.

Problem 1. Find lim x -> 0 (sin(x)/x).

Problem 2. Prove that lim x -> 0 (sin(x)/x) = 1.

If Problem 1 were given to a person who hadn't used calculus much
recently, and had forgotten the seminal importance of that limit (**), but
*had remembered* L'Hospital's Rule, that person could quickly and
effectively *find* that limit using L'Hospital's Rule:

lim x -> 0 (sin(x)/x) = lim x -> 0 (cos(x)/1) = 1/1 = 1.

Here, of course we have used d/dx (sin(x)) = cos (x).

So, if all that were asked for is to find (compute) that limit, then
L'Hospital's Rule probably is that person's only hope of getting the
correct answer, and I would say that using it to solve Problem 1 is
effective and correct.

-------

If Problem 2 were given in isolation, with no context given that would
indicate whether L'Hospital's Rule were too advanced or not yet covered in
the course, then I would still say that using L'Hospital's Rule is
effective and correct, although I would expect that some would argue the
contrary.

--------

(**) The limit lim x -> 0 (sin(x)/x) = 1 is used to *derive* the formula
d/dx (sin(x)) = cos (x).

--------

As to the original problem that Karplus and Chapman referred to, since it
is do-able without L'Hospital, one could say that Karplus used an elephant
gun to kill a fly, but that the shooter was NOT blind.

There is a problem very similar to this (page 110, #30) in Stewart's
"Calculus Early Transcendentals" 4th. Ed. This is well in advance of
Stewart's presentation of L'Hospital.

So, I would have to say that Karplus used a high-caliber method (Ka-Boom!).

--- Joe (Sent via 10.2.8 at 12:51pm PDT 10-22-2004).

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