Ways to know whether my Laptop fan is working or not

Solution 1:

You've exhausted what I would do, check CPU core temps then try get info regarding fan RPM. If your CPU Core temps are in the 70s-80's and you can't hear any fans (you would almost certainly hear them) then I think you have a problem.

EDIT: Download SpeedFan, manually ramp up the fan RPM and if you don't hear anything then you've got a problem.

Solution 2:

You can always pick your laptop up (when it is running) and see underneath if you see some obvious things happening e.g. winging sound or some air being fed out.

I am making an assumption that your PC/Laptop is reasonably old (>2yrs?). Depends on what things you have used it on and how much you have used it on average, The thermal layer on the processor die base runs out accordingly. For example, if you had a GPU chip with the processor built-in, using the laptop for gaming will mean your thermal stuff is gonna run out a bit quicker (but not like tomorrow or next month). The solution is to put the new paste (costs about 10 bucks).

P.S. There is this thing about revs for the fan. If your fan is revving @ lower speed (<60), you will never be able to hear it properly anyway. Check Open Hardware Monitor for the tools to check your fan revs (sorry if You have done it already!)