This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition.
Let $A$ be a C*-algebra and let $A^{**}$ be its enveloping von Neumann algebra. Second-dual C*-algebras are discrete, which means they are (weakly) generated by minimal projections.
Q. Is $A^{**}$ generated by projections of the form $w-\lim_{n\to \infty} f^n$, where $f\in A$ is a positive element of norm at most 1?
Yours answer is negative. For example consider the C*-algebra $C[0,1]$. Let us consider the (Borel) charactrisitic function $\chi_{\{Q^c\cap[0,1]\}}$ (which is in the second dual $C[0,1]^{**}$). By your formula this projection does not obtain.