I am currently reading John S. Townsend's "A Modern Approach to Quantum Mechanics." In section 2.2 he introduces the $\hat J$ operator, which he refers to as "the generator of rotations." He gives the following equation, which expresses an infinitesimal rotation about the z-axis in terms of $\hat J$ : $$\hat R(d\phi \boldsymbol{k})=1-{i \over \hbar}\hat J_zd\phi$$ He then goes on to explain that we can build any rotation needed by using an infinite number of these infinitesimal rotations. Thus: $$d\phi=\lim_{N \to \infty}{\phi \over N} $$ Using these two expressions, he writes a third expression which is: $$ \hat R(\phi \boldsymbol{k})=\lim_{N \to \infty}\big [1-{i \over \hbar}\hat J_z\big({\phi \over N}\big)\big ]^N$$ I understand everything except the $N$ in the exponent of the final expression. I've considered the idea that writing the $N$ in the expression is just a formalism. Because when the rotation operator is expressed as above it can also be written like this: $$\hat R(\phi \boldsymbol{k})=\exp\big [{-i\hat J_z\phi \over \hbar}\big]$$ The idea that it is simply formalism seems wrong to me, however, because if this were true why not just define $\hat R(d\phi\boldsymbol{k})$ differently in the original expression so that $N$ would appear as desired in the final expression? This leads me to believe there is some mathematical step I am missing.
What exactly is the source of $N$ in the exponent?
