The lie bracket as a limit

Question

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see that $$\lim_{t\to0}f^*_t(\frac{(w(x_1,\dots,x_i,(f_t)_*x_{i+1},\dots,(f_t)_*x_k)-w(x_1,\dots,x_{i+1},(f_t)_*x_{i+2},\dots,(f_t)_*x_k)}{t})=w(x_1,\dots,[-X,x_i],\dots,x_k)$$