Expert: Paul Klarreich - 10/31/2008

Question
Dear Sir :
Can you show me the value of the Lim (sin(x)- (x))/x^3
as x approaches zero
The idea is not to use L'hopital
Thank you in advance for your help

Answer
Questioner:   David
Category:  Calculus
Private:  No
 
Subject:  limits
Question:  Dear Sir :
Can you show me the value of the Lim (sin(x)- (x))/x^3
as x approaches zero
The idea is not to use L'hopital
Thank you in advance for your help
................................
Hi, David,

   sin x - x
lim ----------
      x^3

I am not sure this is appropriate, but right now, this is all I have for you:

The power series expansion of sin x:

sin x = x - x^3/3! + x^5/5! - ....

Then

sin x - x =  - x^3/3! + x^5/5! - ....

and

(sin x - x)/x^3 =  - 1/3! + x^2/5! - ....

and the right side will approach  -1/3!, or  -1/6, which should be the answer.

If I can come up with something better, I'll let you know.