# How do I do this math problem?

Discussion in 'Odds & Ends' started by Manus Domini, Oct 22, 2011.

1. ### Manus DominiHall of Fame

Joined:
Jan 10, 2010
Messages:
4,809
Location:
Jersey
Before I give the problem, I want posters to know I am NOT asking for the answer, so don't give it to me. Nor should you give away the entire process, because all I have to then do is plug in numbers, so you are giving me the answer anyway. I just want where to start and to be let known if I am doing the problem incorrectly.

So, the problem I am assigned is:

"analytically simplify the following limit, which represents the definition of the derivative f'(a) for the function f(x)=sinx and x=a."

My teacher doesn't give the limit, and so I don't know where to start. The substitute (since the teacher wasn't there yesterday) said something about about the "angle addition" rule or something, so what I have so far is:

(sin(x+dx)-sinx)/(dx)-->(sinxcosdx+sindxcosx-sinx)/(dx)-->(sinx-sinx+sindxcosx)/(dx)-->(sindxcosx)/(dx)-->(sin0cosx)/0=limit does not exist

but the limit should definitely exist at all times except when sinx-->0, which it isn't necessarily doing.

2. ### Hot SauceHall of Fame

Joined:
Aug 23, 2006
Messages:
1,882
Location:
Vancouver, BC
Dunno.....

3. ### ClaudiusProfessional

Joined:
Jun 4, 2009
Messages:
1,031
The limit as dx approaches 0 exists, and is equal to the derivative of sin(x), which is cos(x).

To simplify, you will need to use these two facts:

1.) lim x--> 0 sin(x)/x = 1
2.) sin(x) - sin(y) = 2cos(1/2(x + y))sin(1/2(x-y))

So,

lim dx --> 0 (sin(x+dx) - sin(x))/dx = lim dx--> 0 ((2cos(x + dx/2)sin(dx/2))/dx = lim dx --> 0 cos(x+dx/2)sin(dx/2)/(dx/2) = cos(x + 0/2) * 1 = cos(x).

Hence, d/dx (sin(x)) = cos(x).

4. ### TammoBanned

Joined:
May 23, 2011
Messages:
3,866
is this Calculus?

5. ### Slayer_of_KingsNew User

Joined:
Jan 27, 2007
Messages:
1
It's good 'ol Trig.

6. ### FrankenstineNew User

Joined:
Aug 23, 2011
Messages:
91
No. It's definitely calculus.

7. ### The BawssBanned

Joined:
May 8, 2011
Messages:
2,826
Location:
Lyon, France.
Yes. 10truths

8. ### PolarisHall of Fame

Joined:
Mar 28, 2005
Messages:
2,319
I would have said, "You're on the right track, but your evaluation of the limit of (sin x)/x needs more attention."

Instead, Manus now has the answer in its full glory, which he did not want .

Last edited: Oct 23, 2011
9. ### sureshsBionic Poster

Joined:
Oct 1, 2005
Messages:
43,223
This question is extensively discussed on the Web, including the fact that the proof of the limit of sin(x)/x as x->0 is 1 should come from some method (like geometric analysis) rather than L'Hospital's rule, otherwise it will become a circular proof.