Non-colliding Brownian particles in one dimension is studied. $N$ Brownian particles start from the origin at time 0 and then they do not collide with each other until finite time $T$. We derive the determinantal expressions for the multitime correlation functions using the self-dual quaternion matrices. We consider the scaling limit of the infinite particles $N \to \infty$ and the infinite time interval $T \to \infty$. Depending on the scaling, two limit theorems are proved for the multitime correlation functions, which may define temporally inhomogeneous infinite particle systems.