what is the limit as x approaches 0 of sinx(1-cosx)/-2x^6

Question

i can not figure out whether the the limit is 2.5E14 and -2.5E14 and therefore, Dne.  The limit could be 0 or it could be something else.

Answer 1

Brian,

 It looks like this is the expression you are taking the limit of:

 

  sin(x)(1-cos(x))

  ----------------

       -2x⁶

 

If so, then as x→0 we have:

  sin(x) --> x

  (1 - cos(x)) --> (1 - (1 - x²/2))

                   --> x²/2  

 

So the limit would be the same as this expression:

    x * x²/2

   ----------     =  x³ / (-4x⁶) = 1/(-4x³)

     -2x⁶

 

As x→0 this will go to infinity, that is, it doesn't have a limit.