how to check of RAM Slot is broken

After all is said and done... with no insult intended or malice of forethought...

So once again, how could I check if the memory slot is broken and in case it is how could I fix it?

You apparently checked both of your sticks of Ram in the one working slot, and the laptop started and worked with 2gb each time you did this. Just so you understand me, I'm saying that you put one stick of Ram in the good slot, turned the notebook on, saw that it worked fine... then you removed that stick of Ram, put the other stick in the same slot, turned the notebook on and saw that it Ran fine. That IS what you did, right? Because if you did not follow that procedure, then we don't know what is wrong with your computer yet... and we won't know until you actually do that.

RAM stick A in slot 0

RAM stick A in slot 1

RAM stick B in slot 0

RAM stick B in slot 1

Why?

You need to be thorough. That's why. Because it is the difference between purchasing a replacement stick of Ram, and having to purchase a new logicboard/mainboard. That is a BIG difference. Yours would not be the first Macbook with a bad ram slot.

This is how you can check to see if the slot is malfunctioning. However, as far as how you can repair it, you can't. You can replace the logicboard/mainboard/motherboard. But you can't somehow detach the slot itself and replace it with another one. The issue might not even be with the actual slot itself.