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I'm trying to understand the derivation in my textbook of the wave function for a potential step. The derivation reaches the step shown in the attached photo, which I am fine with.

However, the book then says:

One boundary condition is that the wave function ψ2(x) must remain finite, which means that the coefficient B2=0.

Why can't we use this same reasoning to eliminate B1 as well? Clearly the B1 term increases without bound as x approaches negative infinity, exactly as the B2 term does as x approaches positive infinity.

And yet, B1 is later solved in terms of k1, k2, and A1. Any thoughts?