How is the wavefunction within a rectangular potential barrier an evanescent wave?

I thought an evanescent wave was exponentially decaying. If you take x->infinity for the following function for E<V, one of the terms goes to 0 but the other term goes to infinity, so the wavefunction will go to infinity. Thus, it isn't physically realizable. But how you get the physically realizable wavefunction for E<V, what does it look like, and how is it an evanescent wave?

http://en.wikipedia.org/wiki/Schrödinger_equation#Time_independent_equation

I thought an evanescent wave was exponentially decaying. If you take x->infinity for the following function for E<V, one of the terms goes to 0 but the other term goes to infinity, so the wavefunction will go to infinity. Thus, it isn't physically realizable. But how you get the physically realizable wavefunction for E<V, what does it look like, and how is it an evanescent wave?

http://en.wikipedia.org/wiki/Schrödinger_equation#Time_independent_equation

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