How to tell if my fan is broken...

Starting at the beginning, I was having issues with my laptop overheating, so I bought a can of compressed air and got rid of a bunch of dust that had built up. This helped, but not completely, so I sprayed into the fan vent again while the computer was on. The fan stopped immediately, which I now realize was the fan breaking, not just the temperature dropping rapidly.

Without my fan working I'm running at an average of 70 C, which is a problem as it creeps up to around 80 and then dies if I stress it out too much.

My question is, is there a way to diagnostically check if my FAN is broken or if I messed something else up? Because a new fan isn't that expensive, but I don't want to replace it if it's not going to work once installed.