Because this limit exists we can invent a new function symbol representing an infinitely tall and infinitely narrow pulse. We define as an infinitesimal pulse of area 1 at and zero everywhere else. This is called an "impulse". Mathematically, introducing impulses is a big step, because does not satisfy the usual requirements for a function. It is a a distribution, which generalizes the notion of a function. But we do not expect you to learn that very advanced material for use in this class. Notice what the impulse did: it "instantaneously" changed the current through the inductor, thus setting the initial conditions for the exponential decay that follows. In fact, it incremented the current in the inductor by the area of the impulse divided by the time constant of the circuit, as seen from the inductor