there is a limit for that. As you said that you could erase what you have surfed for on your computer only but you might not be able to raise about your whole experience on the Internet that has been stored in the main server and that server is your Internet service provider.
These Internet geeks and hackers etc... They are able to do anything with the server having any kind of high profile firewall or security system , so they could erase their privacy easily from the main servers.