L'Hopital's Rule

[Zen]

L'Hopital's rule is used in calculus to evaluate limits that have an indeterminate form. As such, successive differentiation of the numerator and denominator could reveal the exact value of a limit that was apparently indeterminate.

Might be a real geeky pun at work here, but there's the beauty of it all - within the apparent complexity lies the rather simple answer, as is the case with the rule's application.

http://mathworld.wolfram.com/LHospitalsRule.html

Well, I might have lost something through some fault of my own, but the principle holds true - everything goes both ways in the end.

[Ikaru]

Calculus was, uh... not my best subject, but I do remember going over this in AP Calc in 11th grade...slightly.

It's pretty interesting, though, the way you apply it to a general situation...

Although a lot of the times, it IS the case, I think when you say:

within the apparent complexity lies the rather simple answer

I don't think it's as much "simple" answer (note, singular) as it is a number of smaller steps that lead to the same result. That's how I take care of a lot of things (not always a good thing); instead of trying to do things the obvious way when it's not working out, try to find another method that might be faster, or easier, or less tedious.

...and I guess that's more or less what you said.

This is just me trying to sound smart; I DO remember going over this but I was really, really, bad at calc @.@

I was much better off in AP Stats the next year <3

[Zen]

I don't think it's as much "simple" answer (note, singular) as it is a number of smaller steps that lead to the same result. That's how I take care of a lot of things (not always a good thing); instead of trying to do things the obvious way when it's not working out, try to find another method that might be faster, or easier, or less tedious.

Aha, the process by which L'Hopital's rule is applied is differentiation - so if one does not have the ability to differentiate, one cannot apply the rule to solve the problem

Technically a touch more complex than direct substitution or eliminating like terms, but it's still interesting - had a massive argument with someone that led to a falling out, and the formula came to mind somehow.

Calculus probably gets more hatred than it is due, but I don't blame those who dislike it - integration gets on my nerves sometimes XD

[Zen]

Haha, it's nice to see someone trying new stuff out in maths! However, the rule is actually simpler than your calculator method - the explanations given are thorough to the point that they make it seem complex.

The simplified explanation for L'Hopital's rule is as follows:

- Assume you have a function in the form [ f(x) / g(x) ]

- The limit has x tending to zero ( x -> 0 )

If the individual equations f(0) and g(0) both equate to zero, direct substitution yields 0/0 for the limit - this is an indeterminate form.

So, if the derivatives of f(x) and g(x) are f'(x) and g'(x), respectively:

lim ( x -> 0 ) [ f(x) / g(x) ] = lim (x -> 0 ) [ f'(x) / g'(x) ]

Repeat the differentiation of f(x) and g(x) until direction substitution of x yields a definite value

For example, if the limit is in the form lim (x -> 0 ) [(sin x)/x]

(sin 0)/0 = 0/0

It is an indeterminate form, so the rule is applicable.

Let f(x) = sin x, g(x) = x

f'(x) = cos x, g'(x) = 1

So the limit is now lim ( x -> 0 ) [ (cos x)/1 ] = cos (0) = 1.

if direct substitution gives you zero over zero, then change the limit to have the derivatives of the given functions in their respective places, and substitute

... And to think all this started with a codified rant. Wut.

[Zen]

Not necessarily 1. The end result can be any definite value.

It just needs to start at 0/0.

[Maelstrom]

It's been years since I took my only college math class (calc 2) and a couple years before then since I'd taken calc 1, so really don't remember much except the name of the rule. You don't use calculus in biology. Advanced math and stats, yes, but not calc.

[Zen]

You could actually use differential equations for population studies. Can't recall the exact equation, but I think it had something to do with carrying capacity of a habitat.

Hmm, come to think of it, calculus is a busybody that sticks its fingers into all the branches of maths.