Math Is Fun Forum

Daramarak

Guest

I just want a hint (Limits & Calculus)

I am banging my head, and I cannot stop thinking of it.

lim x->0 ((e^x - 1 -x)/(x sin(x)))

A hint on how to crack this would be nice...

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: I just want a hint (Limits & Calculus)

I don't know how much of a hint you want.. so here is a vague one:
You must apply *twice* a rule developed by a certain French mathematician.

Daramarak

Guest

Re: I just want a hint (Limits & Calculus)

Le Hôpitals rule twice?

I have never though of applying it twice, but I can see that both the first equation and the second satisfies the conditions...

Hmm, I have to get my pencil now...

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: I just want a hint (Limits & Calculus)

Yep, you can keep applying it any number of times, as long as the expression is an indeterminate form !

Daramarak

Member

Registered: 2006-11-01

Posts: 1

Re: I just want a hint (Limits & Calculus)

Yeah, it worked like a charm. It is nothing as peace of mind

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: I just want a hint (Limits & Calculus)

Well, but I am a little bit confused, is the final result 1/2 ???

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X'(y-Xβ)=0

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: I just want a hint (Limits & Calculus)

Well, but I am a little bit confused, is the final result 1/2 ???

Indeed it is.

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: I just want a hint (Limits & Calculus)

Actually, the rule was developed by a Swiss person, who was paid off by a French person, or so is believed

http://en.wikipedia.org/wiki/L%27hopitals_rule

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: I just want a hint (Limits & Calculus)

Actually, I would also recommed the Taylor Expansion approach for this problem. I think it's more intuitive, plus it has the advantage of also being useful in situations where you aren't dealing with 0/0 or inf/inf, but you want to simplify the function without removing the x dependence.

You know:
[align=center]

[/align]
and
[align=center]

[/align]

As can be shown with a Taylor Expansion. When you take the limit as x becomes small, you know the higher order terms become small faster than the lower order terms, so you can ignore terms above x^n to the "nth order" of accuracy.

Expanding to first order still leaves you with 0/0 for this problem (which is why you need two iterations of L'Hopital's Rule), but to second order,

[align=center]

[/align]

In the limit as x->0, which gives you the same answer.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: I just want a hint (Limits & Calculus)

Well, in the end I cannot help but treat sinx and xcosx as x+x+o(x)

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X'(y-Xβ)=0

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: I just want a hint (Limits & Calculus)

I assume you are talking about taking the derivative of the denominator in using L'Hopital's Rule here. Yes, it is definitely true that you can make the simplification there that sin(x) + xcos(x) = x + x + O(x^2) for x very close to zero ... not quite sure where you're going with this though ...

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: I just want a hint (Limits & Calculus)

Having using L'Hopital's Rule Once

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X'(y-Xβ)=0