The lifecycle of a servlet if managed by the servlet container and not by CDI. However, CDI injection is expected to work in servlets.
A servlet container will also provide some built-in beans that can be injected using CDI:
A servlet container must provide the following built-in beans, all of
which have qualifier @Default:
a bean with bean type javax.servlet.http.HttpServletRequest, allowing
injection of a reference to the HttpServletRequest
a bean with bean type javax.servlet.http.HttpSession, allowing
injection of a reference to the HttpSession
a bean with bean type javax.servlet.ServletContext, allowing injection
of a reference to the ServletContext
If you need to inject a servlet somewhere, you are probably doing something wrong.