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The reason for the change seems quite clear to me- the free ride is over. Just like what happened with routing at the end of last year, ESRI wants to now lock you into paying for these services using ArcGIS Online accounts & credits.
ESRI still provides two elevation globeserver REST endpoints but probably not at the resolution you want:
http://services.arcgisonline.com/arcgis/rest/services/Elevation
Looks like you'll either have to a.) pay for credits & use ESRI's previously free service, b.) load up and serve your own elevation data, or c.) hope that the USGS or other fed agency is already serving up elevation data as a REST service someplace.
tammy,
apparently the appropriate replacement service can actually be found here.
its my understanding that even though the service requires an ArcGIS Online subscription (which is now included with the purchase of ArcGIS Desktop automatically) it doesn't actually burn credits.
as a developer, what this means to you is that you'll have to authenticate on behalf of end users to access the content if you don't want anyone to have to sign in. this would entail setting up our new resource-proxy and either registering your app or storing hardcoded user credentials and routing traffic to the elevation service through your proxy.
this is similar to what you'd do if you were interested in using other Esri services like traffic, batch geocoding or routing.
actually, it looks like the identifyParameters for that image service are a fair amount different than what we see for map services.
i took the time to write up a sample which uses esriRequest to gather the info you're looking for. you'll have to sign in when running it in jsfiddle, but the same code could utilize a proxy on your own web server.
let me know if i can be of any more help.