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L'Hospital is NOT Blind
Posted:
Oct 22, 2004 4:36 PM


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 doable 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 highcaliber method (KaBoom!).
 Joe (Sent via 10.2.8 at 12:51pm PDT 10222004).
 Delete the second "o" to email me.
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