Let $G$ be a connected real Lie group and $\mathfrak{g}$ its real Lie algebra equipped with some inner product and the corresponding norm $\|\cdot\|$. Let $d$ be the left-invariant Riemannian distance on $G$ defined using the norm $\|\cdot\|$.
As far as I know, it is not always true that one parameter subgroups $\exp(tX)$, for $X\in\mathfrak{g}$, are geodesic lines in $d$. However, are they locally geodesic? Or rather something weaker, fixing $X\in\mathfrak{g}$, do we have $$\lim_{t\to 0} \frac{t\|X\|}{d(exp(tX),e_G)}=1?$$