There are two ways to get the extent of a route, one with a map and one without.
With a map:
FindRoute with RouteFinderOptions.returnMap = True RouteInfo returns mapImageInfo, which will have an envelope object.
Without a map:
GetRoute with RouteFinderOptions.returnDirection = True ; and RouteFinderOptions.RouteOptions.returnGeometry = true ; RouteInfo will return segDescription and totalDescription. segDescription has an envelope for each segment, and totalDescription has an envelope for the total direction.