Let G be a Lie group, let L(G) be its Lie algebra, and let exp : $L(G){ ightarrow}G$ denote the exponential mapping. For $S{subseteq}G$, we define the tangent set of S by $L(S);=;{X;{in};L(G);:;exp(tX);in;S;for;all;t;{geq};0}$. We say that a semigroup S is strictly infinitesimally generated if S is the same as the semigroup generated by exp(L(S)). We find a tangent set of the semigroup of all non-singular totally positive matrices and show that the semigroup is strictly infinitesimally generated by the tangent set of the semigroup. This generalizes the familiar relationships between connected Lie subgroups of G and their Lie algebras