The first circuit is for \$t\$ (time) \$\le 0\$, and the second is for \$t \ge 0\$.
The book I'm reading says that \$i_L(0^+)=i_L(0^-)\$, but \$i_1(0^-) \neq i_1(0^+)\$ because there are no limitations on the current changing instantaneously at \$t_0\$.
What I don't understand is why \$i_L(0^+)=i_L(0^-)\$, but \$i_1(0^-) \neq i_1(0^+)\$? Wouldn't \$i_L(t)\$ change instantaneously as to whether or not the voltage source would be connected? Why is one instantaneously changeable and not the other?
Best Answer
We are talking about the mathematical models used to model a lumped-element network.
In this context any current can change instantaneously, unless there is a reason it can't. The reason why \$i_L\$ can't change instantaneously is that that current flows in an inductor.
Current flowing in an inductor cannot change instantaneously. Any attempt to force such a current to change instantaneously would generate a voltage spike of infinite amplitude across the inductor (a Dirac's delta pulse, from a mathematical POV).
In practice, trying to change a current in an inductor abruptly will lead to large voltage spikes (not infinite) until some nonlinear phenomenon will arise in the circuit (e.g. arcing) and will quench the spike.