Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

sin(x)/x

Ok, so:

I can show this l'Hôpital's rule:

But here's my question. When writing out the derivation for:

you have to at some point use the fact the lim (x->0) sin(x)/x = 1. I can't prove that it's true with out using l'Hôpital's rule. l'Hôpital's rule requires the use of the derivative of sin(x). I'm not sure if I'm making myself clear, but this seems to me like circular reasoning. Can anyone show me a proof of:

That does NOT use l'Hôpital's rule? I can do it graphically, but somehow that doesn't seem like it's good enough...

P.S. Bonus points for anyone who can tell me how to pronounce l'Hôpital by spelling it phonetically.

Last edited by bossk171 (2007-09-07 06:25:00)

---

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: sin(x)/x

"Low-pee-tal"

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: sin(x)/x

Thanks! Which syllable has the emphasis? LOW-pee-tal, low-PEE-tal or low-pee-TAL?

---

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: sin(x)/x

at my school they always said 'low pea tall's rule.

and yeah, we couldn't use l'hopitals rule to find the limit of sin(x)/x while trying to find the derivative of sin(x), cause we'd need to know it before hand.

so yes it can be proven without it. I think the only one I've seen uses geometry and the squeeze theorem.

geometrically its kind of easy to visualize. Look at this rough diagram i drew. if x is the angle of the small circular sector i drew (in radians) we know since this is the unit circle, that the length of the small arc it encloses is radius*x, which is just x. we also know sin(x) is the vertical height from the origin. see how close the length of that arc is to the sine? the reason the limit turns out that way is a very very very small portion of an arc is virtually the same as a line.

thats not a proof of course, just a nice way to visualize it.

Last edited by mikau (2007-09-07 06:54:36)

---

A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: sin(x)/x

Emphasis goes on the "PEE".

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: sin(x)/x

yes, but i think what boski is asking is how do we find lim sin(h)/h if we don't know its derivative and thus, can't use l'hopitals rule.

---

A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: sin(x)/x

Ah, I got a bit mixed up.  Here is a proof of this:

http://people.hofstra.edu/Stefan_waner/ … iglim.html

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: sin(x)/x

Exactly mikau, thanks for the picture.

Very helpful Ricky.

THANKS!

(thats (THANKS)(THANKS-1)(THANKS-2)...(3)(2)(1) by the way)

Last edited by bossk171 (2007-09-07 07:05:22)

---

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: sin(x)/x

ah there it is. It seems like thats the only proof anyone ever uses. I wonder if there are others.

I mean, sine is a geometric function so it makes sense to exploit its chracteristics through geometry. But i wonder if there are other ways.

---

A logarithm is just a misspelled algorithm.