My car recently started overheating real bad, so I checked things out. I noticed only the left radiator fan is going all the time, and I also found a large split in the overfill tank, which is leaking water.

Someone told me that the radiator fan that isn't going all the time is designed to click on only under certain circumstances.

Is this true? If so, under what circumstances?

*NOTE*
I checked the old posts on it. Some said the AC turns it on, but I turned my AC on and it did nothing.
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1996 Eclipse RS

do you mean the left side of the car or left side when your looking into the engine bay? ....Looking into the engine bay the fan on the right is the fan that goes on for the AC. The one on the left is to cool the engine witch only turns on after the car reaches "normal" temps. If you wanna see if its working then drive around and heat up the engine nicely and turn off the A/C then park and pop the hood. One of the fans should be on(the one to your right). If not then you have a problem. When the A/C goes on the one to your left should turn on. If one of them dont work it could be a relay, the fan motor etc... I changes the motor on my engine fan recently and i have the relays linked for some double cooling action. hehe. hope this helps you.

*edit* and if you can patch up the hole in your overflow tank then do it. loosing coolant is not a good thing...