The formula you want, in terms of the variables you have defined, is:

To prove this statement requires some methods usually found in a second semester of elementary calculus. To begin, let's write:

Because the constant does not depend on , we may divide through by this to obtain:

Now, taking the natural log of both sides, we have:

The log of a limit is equal to the limit of a log, hence:

Applying the property of logs we have:

Dividing through by :

Bring to the denominator:

We now have the indeterminate form 0/0, thus application of L'Hôpital's rule reveals:

Simplify:

We see now the term involving vanishes to zero, hence:

Solve for