When a car has a dying battery - it can't hold a full charge and cannot deliver enough power - it is fairly obvious.
When the starter solenoid goes "CLICK-CLICK-CLICK" that is a sign that there is enough voltage to pull the starter solenoid in, but applying a load to the starter motor drops the voltage below what is required to hold the solenoid in, so the solenoid lets go...which returns the voltage to enough to pull the solenoid in...which drops the voltage... You get the idea.
When the starter struggles to turn the motor over, the battery is close to death, especially when it intermediately clicks dropping the solenoid out. At this stage leaving the battery on a few amps recharging overnight can perk it up somewhat, but you are simply delaying the inevitable. Replace your battery.
But there is another sound/response you will hear which isn't the battery, it is the starter. Give aways are the solenoid pulls in no problems, the starter will crank the engine but very slowly, and then maybe starts cranking a bit faster. Or the solenoid will pull in and she either doesn't crank or starts to crank and then stops. If the battery is nearly dead the solenoid will click drop out, because the voltage has gone too low trying to crank the engine.
A few years ago my Falcon failed to start, and I braced myself to change the starter. But it turned out that the electrical connection had come loose. Phew.
But then, after 13 years of trouble free starting it was again proving difficult. I couldn't quite nail it down - was it the battery? Was it the starter? Some days I thought it was the battery, some days the starter. Was not easy to diagnose. So, I did nothing for a few months. Until a very hot day when it cranked for a bit, and then stopped, never to start again. The battery was pulling the solenoid in just fine, but the motor was doing nothing. Argh. And it's a very hot day. By the way, you can't bump start an automatic Falcon.
Suffice to say a mate of mine towed me home, and first thing the next day I went and sourced a new starter. My mate said to head over to Ashdown-Ingram in Wangara WA, they would look after me. He was correct. These days starters are throw away, no longer exchange, RRP on an after market starter was AUD$170(2019) but they gave it to me wholesale $92. Thanks! Note that some models of Falcon have 9 teeth on the starter, some have 10. Mark one tooth, and then count.
Removing the old and installing the new - it is possible to do without say taking the engine out, but it is hard. You jack up the passenger side of the car, and go under. There are two bolts holding it in. One at the bottom which is easy enough to get to. One at the top which is not easy to get to, and you can't see it. You have to go by feel. I had a socket on a 20cm extension and I just kept poking around in the general area until it felt like I was on the bolt. Then I plugged in the ratchet, carefully, and undid/redid. You will get frustrated. You will lose skin. But you can do it. Years ago I put a little mirror on a bendy piece of wire, I used this to see behind the starter to get my bearings, and tried again. Stay calm, be patient, you'll make it.
Once you have your new starter installed you won't believe how quickly the engine turns over! You don't realise how slow it becomes until you compare it to a new one!