use Kirchoff's Law for a series RL circuit, $E - V_R - V_L = 0$

$E - IR - L \cdot \dfrac{dI}{dt} = 0$

$E - IR = L \cdot \dfrac{dI}{dt}$

$\dfrac{dt}{L} = \dfrac{dI}{E-IR}$

$- \dfrac{R}{L} \, dt = -\dfrac{R}{E-IR} \, dI$

$-\dfrac{Rt}{L} + C = \ln|E-IR|$

at $t=0$, $I = 0$ $\implies C = \ln|E|$

$-\dfrac{Rt}{L} = \ln|E-IR| - \ln|E| = \ln \bigg|\dfrac{E-IR}{E} \bigg|$

$\dfrac{E-IR}{E} = e^{-Rt/L}$

$I = \dfrac{E(1-e^{-Rt/L})}{R}$

$\displaystyle \lim_{t \to \infty} \dfrac{E(1-e^{-Rt/L})}{R} = \dfrac{E}{R}$