The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δ r by Δ r / N and Δ t by Δ t / N, where N is a positive non-zero integer. Then as N increases, the magnitude of Δ r and Δ t become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators.