why do membrane potentials become more positive when positive current is injected?

Answer 1

What happens is that the current causes Sodium channels to open for a short while, allowing Na to diffuse back into the cell and lower the membrane potential. However, the current must not be enough to activate the action potential.

Answer 2

Finally a question I'm perfectly trained to answer.

I'm an electrophysiologist, so I regularly inject current into cells to make them more positive. The reason is because cell membranes are resistors. Due to ohms law, V=I x R. The current times the resistance of the cell membrane equals the voltage generated. Classically, neurons have a membrane resistance of between 100 and 1000 megaOhms. Currents injected are usually lower than 1nA; though in certain occasions more is needed.

Of course, as the person mentioned above, if you inject a lot of current you can make certain cells undergo a chain reaction of molecular events that is the action potential; but that is not needed for the simple Ohmic relationship between current, voltage and resistance.