| When your sprockets no longer have nice teeth on them and are worn thin or even hooked you'll know they're knackered.
When you need to adjust your chain after every ride and adjust it so that it flexes 25-30mm at it's tightest spot but it hangs down like a skipping rope at it's slackest spot you'll know the chain is knackered.
You may even get nasty graunchy noises or the chain skipping teeth.
How long they last is a bit like asking how long is a piece of string. If you live on a beach, get sand in your chain and never clean or oil it - probably 1000 miles. If you clean and oil it regularly, adjust it properly, ride smoothly so that you're not giving the chain a hammering by chopping on and off the throttle then 15K would be about average.
A worn sprocket can knacker a chain quickly and a knackered chain can wear a sprocket in no time so change them all together when you do it | 
Jools
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