I hope I am not posting too many questions in a row regarding matrix exponential, but I am solving exercises and got stuck trying to prove $$B(u) = \lim_{t\to0^{+}}\frac{e^{tB}u - u}{t}$$ for $B \in M(d \times d, \mathbb{R})$ and $u \in \mathbb{R}^d$. I think it's related to eigenvectors (or generalized eigenvectors) of $B$, I also tried the various definition of $e^{tB}$, i.e as limit, power-series, etc but didn't get anywhere.

Hints?

Power-series:

$e^{tB}u - u = \sum_{k=0}^{\infty}\frac{t^k}{k!} (B^k - id)(u)$