JavaScript (Node.js), 23 bytes
(a,b,i)=>a*10n**i/b%10n
JavaScript (Node.js), 43 bytes, port
(a,b,i)=>eval(`for(;a=10*(a%b),i--;)a/b|0`)
JavaScript (Node.js), 51 bytes
(n,d,x)=>[...Array(x)].map(_=>[n*=10,u=n/d,n%=d])|u
use Array rather than recursive to avoid stackoverflow