I need to show that for every $p\in M$ and $v,w \in TpM$, we have
$$ \lim_{t\rightarrow \infty } \frac{d_{g}(\exp_{p}(tv),\exp_{p}(tw))}{t}=|v-w|_{g}$$
is a problem of the book "introduction to Riemannian Manifolds"" by J.M.Lee and he gives a hint:Use the Taylor derires of $g$ in Riemannian normal coordinates on convex geodesic ball centred at p.
$$ g_{ij}=\delta_{ij} -\frac{1}{3}\sum_{k}R_{iklj}x^{k}x^{l}+O(|x|^{3})$$
I tried to use this hint but I can't solve, on the other hand, the first edition of the book gives another hint: If $\epsilon$ is small enough and $X,Y \in \nu $, show that there exists a constant $c > 0 $ such that
$$ (1-c|t|)|tX-tY|_{g}\leq d_{g}(\exp_{p}(tX),\exp_{p}(tY))\leq (1+c|t|)|tX-tY|_{g} $$
whenever $|t|\leq 1 $ by comparing g with the Euclidean metric in normal coordinates and using this result, conclude that
$$ \lim_{t\rightarrow \infty } \frac{d_{g}(\exp_{p}(tX),\exp_{p}(tY))^{2}}{t}=|X-Y|_{g}^{2}=|X|_{g}^{2}+|Y|_{g}^{2}-2<X,Y>_{g} $$
until the moment I could demonstrate the two hints but I need to apply them correctly because I have not concluded what I need.
Any suggestions or help would be appreciated