Booth defines the infinitesimal generator of a lie group (denote the manifold it defines by $M$) using flow $\theta_t(p)$ by calculatng the limit (mainly the derivation for $f$ in each point $p\in M$) $$X_pf=\lim_{\Delta t \to 0}\frac{f(\theta_{\Delta t}(p))-f(p)}{\Delta t}.$$ However in these lecture notes , the matrix form of the generator is defined using the taylor expansion of $\theta$ (page 8). The limit above looks a bit like a derivative but still I can't understand the equivalence between the definitions.
Why these both representations are equivalent? Can one prove it ?