When a route fails and has no feasible successor, EIGRPuses a distributed algorithm called Diffusing Update Algorithm (DUAL). DUAL sends queries looking for a loop-free route to the subnet in question. When the new route is found, DUAL adds it to the routing table. The EIGRP DUAL process simply uses messages to confirm that a route exists, and would not create a loop, before deciding to replace a failed route with an alternative route. A router would send an EIGRP query message to its working neighbors. A router would respond with an EIGRP reply message, stating the details of the working route to the subnet and confirming that it is viable, if it has a route that works, anyway.
Replacing a failed route with a feasible successor takes a very short time, typically less than a second or two. When queries and replies are required, convergence can take slightly longer, but in most networks, convergence can still occur in less than 10 seconds.