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So I have bought a bike but the easiest gear on the cassette is a 30 tooth and the derailleur is a short cage derailleur, the chain keeps falling off this gear and back to the next easiest gear, is this the limit screws and if so which one, or is it the short derailleur and if so should I get a derailleur hanger extender?

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    Which derailleur do you have? Is the pulley wheel clearing the cassette properly? Are you sure it’s not just a matter of adjusting the limit screws and cable tension (and maybe B screw)?
    – Michael
    Mar 28, 2021 at 18:24
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    Brand new bikes often suffer from a short "settling" process as all the cables stretch and relax into the stops. Its common for a shop to offer a free tuneup at about 6 weeks to tweak all the gearing. I'd suggest tuning it up before buying new parts.
    – Criggie
    Mar 28, 2021 at 18:43

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Firstly, like what @Michael has said, I recommend dialling in the settings for the derailleur first.
The limit screws can be dialled in easily, and there are numerous guides online, especially from Park Tool.
The B screw is one that you need to play around with too. You need to ensure that the guide pulley wheel actually clears your 30tooth sprocket, and have at least 3mm of space in between.

Secondly, like what @Criggie has said, because it's a new bike, there's sort of a breaking in period for the cables. New cables tend to get "stretched" in the first few rides, therefore losing tension. The loss in tension might be what causes the derailleur to not be able to push the chain up to the 30tooth.
It's easy to re-tension your cable, and there are guides online, such as those from Park Tool as well.
Typically, you can simply turn the barrel adjuster anti-clockwise to increase the tension. But that is for micro-adjustment. If there's a need for heavier tension, it can only be done by undo-ing the cable pinch bolt, and pulling the cable, then pinching the cable again.

Thirdly, is your derailleur hanger aligned? A good way to check is to backpedal. If the chain falls off to a different sprocket when back-pedalling, then it might mean that the hanger is bent. A bent hanger is common, from crash or just bumps onto the derailleur itself. It wreckeds up the indexing of the derailleur, and it could be why the derailleur cannot clear the 30tooth sprocket.
It's easy to fix a bent hanger, just use a big adjustable spanner to bent it back into place (at your own discretion). Or else get those hanger alignment tools such as the Park Tool DAG-3 to bent it back. Otherwise get a new hanger.

Lastly, on the topic of new hanger, a hanger extender might not be the best solution to clear bigger sprockets. I don't know the make of your bike or the hanger compatibility, but a rule of thumb is that the derailleur cage should be as short as possible to accommodate the biggest sprocket you are going to use to avoid compromising shifting performance.
For example, the Shimano 105 R7000 derailleurs. It comes in 2 sizes, the R7000-SS short cage for clearance up to 30teeth, and the R7000-GS medium cage for clearance up to 34teeth. If I am using a 28teeth sprocket, I will choose the short cage version as if the cage is too long the shifting might be compromised.
*but it could just be purely marketing for product proliferation Point is that whatever your bike came spec'ed with, it is probably the spec'ed that works.

Hopefully this helps, I believe that it might be good to play around the settings on the derailleur first!
Nevertheless, a derailleur hanger extender might be a good choice too, because a new bike part a day keeps the crashes away.

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