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Find the limiting value of the current in a circuit

For positive constants E,R, and L define the current in a circuit at time t by

    \[ I(t) = \frac{E}{R} \left( 1 - e^{-\frac{Rt}{L} \right). \]

Determine the value of

    \[ \lim_{R \to 0^+} I(t). \]


We use L’Hopital’s rule to compute the limit as follows,

    \begin{align*}  \lim_{R \to 0^+} I(t) &= \lim_{R \to 0^+} \frac{E - Ee^{-\frac{Rt}{L}}}{R} \\[9pt]  &= \lim_{R \to 0^+} \frac{\frac{Et}{L} e^{-\frac{Rt}{L}}}{1} \\[9pt]  &= \frac{tE}{L}. \end{align*}

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