What happens to the current as the resistor approches infinty?

Most resistors are linear, or 'ohmic', devices -which means that they obey Ohm's Law. So the ratio of voltage to current remains constant for variations in voltage. In other Words, their resistance remains constant -providing their power rating isn't exceeded.

So you can say that, providing the current increase doesn't cause their power rating to be exceeded, their resistance should remain the same. Resistors wouldn't be of much use if their resistance value changed whenever the current through them changed!