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:
$e^{tB}u - u = \sum_{k=0}^{\infty}\frac{t^k}{k!} (B^k - id)(u)$
We have $$e^{tB} - I = \sum_{k=1}^{\infty}\frac{t^k}{k!} B^k$$ (the $k=0$ term vanishes). It follows that $$\frac{e^{tB} - I}{t} = \sum_{k=1}^{\infty}\frac{t^{k-1}}{k!} B^k = \sum_{k=0}^{\infty}\frac{t^{k}}{(k+1)!} B^k$$ We see that $\frac{e^{tB} - I}{t}$ has a removable singularity at $t=0$. Evaulating this series at $t=0$ yields the desired result.