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Let's say, that there is medium, where the speed of light is $10^8 \, \rm m/s$. So in that medium if there exists a particle that is moving through without any hindrance what would be the speed limit of that particle would it $3 \cdot 10^8 \, \rm m/s$ or $10^8 \, \rm m/s$? Since in vacuum the speed limit is equal to the speed of light that is $3 \cdot 10^8 \, \rm m/s$.

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The speed limit is $3 \times 10^8 m/s$. It's possible to be faster than light in a medium. Notably, you'll get Cherenkov radiation in this case.

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