1). Caltrans mainly have freeway, highway. Not doing city streets. You may not get speed limit from caltrans. If it is freeway, you can, but you don't have to. Since CA freeway is 65 miles per hour. Towing truck is 55 miles per hour.
I have confirmed from LA county GIS team, they do not provide speed limit either.
They don't have one way either.
2). Google Speed Limit map is too expensive, NOT a option.
3). Microsoft speed limit is reasonable pricing. But you have to build a tool youself to check speed limit. Microsoft does not provide any map to show you speed limit.
Here is the tool I build, model 31728
https://transparentgov.net/json2tree/esri/server/folder2.html?org=https%3A%2F%2Fpublic.gis.lacounty.gov%2Fpublic%2Frest%2Fservices&url=https%3A%2F%2Fpublic.gis.lacounty.gov%2Fpublic%2Frest%2Fservices&arcgis_online_token=&timeout=9000&select_folder=28&select_layer=0&select_folder_text=LACounty_StreetMap+%3Csup%3EMapServer%3C%2Fsup%3E&select_layer_text=0+%26%23x21E2%3B+Cities+and+Communities+%3Csup%3EFeature+Layer%3Csub%3E+esriGeometryPolygon%3C%2Fsub%3E%3C%2Fsup%3E
4). $0 you can get speed limit from HereMap. But you have to registar with hereMap with a organization ID.
https://demo.support.here.com/examples/v3/pde_speed_limits_in_view