Finite rhymes with 'my time'. Infinite rhymes with 'minute' (as in seconds, minutes, hours and not minute, micro, small).
Why are i's in finite and infinite pronounced differently?
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Finite is pronounced ['faɪnaɪt] while infinite is pronounced [ˈɪnfɪnɪt].
So why is the vowel in the first syllable of 'finite' different from the vowel in the second syllable of 'infinite'?
It's probably because of Trisyllabic Laxing. It's a process whereby a tense vowel (as /aɪ/ in 'finite') is reduced to a lax vowel when two (or more) syllables follow.
And the most common example is pronounce/pronunciation.
There are many different cases of Trisyllabic Laxing. Finite - infinite is one of them.
When we add the prefix in- to the word 'finite', the primary stress from the FI of 'finite' moves to the prefix 'in-'. And now the stressed syllable is followed by two syllables, so the tense vowel /aɪ/ in the root word ('finite', not the prefix 'in-') gets reduced to a lax vowel /ɪ/.
Pronounce/pronunciation is pretty much the same as finite/infinite.