L'Hopital's rule

[sunburnt]

Hi there, I'm currently working on a math project, and I've run into a little snafu -- let me expound
Let me set up the problem

f(x) = cos(x)

f*(x) = lim ( f(x + h) - f(x - h)  )
         h->0          h

Look familiar? Looks like 2 * the symetric difference quotient of f(x).
Here's what I worked it down to:

f*(x) = lim ( -2sin(x)sin(h)  )
         h->0          h

I'm stuck here-- it works out to 0/0. My question is, am I "allowed" to use L'Hopital's rule and instead use the derivative? That was my natural inclination, but I thought one could only use L'Hopital's rule when finding derivatives, which I only sort-of am.

Hope it makes sense :P