In the proof of Gauss's lemma here, there is a step
$\displaystyle\lim_{t\to0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\to0}T_{tv}\ \exp_p(tw_N)=0$
However, the limit seems meaningless (unless tangent bundle has been introduced) since $T_{tv}\ \exp_p(tw_N)$ lies in different tangent spaces for different $t$. Instead I think one can directly conclude that
$\displaystyle\frac{\partial f}{\partial s}(0,0)=T_{0}\ \exp_p(0)=0$
since $f(0,s)$ is a constant curve.
Am I right?
(This step is also present in do Carmo's Riemannian Geometry.)