I have a web project using Resteasy (which in turn uses Weld) and is deployed to Tomcat 7.0.22 (I put the specific version here in case this issue is particular to this version).
I have a ServletContextListener that looks like this:
@WebListener
public class ApplicationInitialisationListener implements ServletContextListener {
    // create a logger here        
    @Inject
    HealthCheck healthCheck;
    @Override
    public void contextInitialized(ServletContextEvent event) {
        if (healthCheck == null) {
            log.error("healthCheck is null");
        }
    }
    @Override
    public void contextDestroyed(ServletContextEvent servletContextEvent) {
    }
}
After deploying to Tomcat, healthCheck is null was logged, and I also noticed this line in the log:
<2013-11-13 13:27:40,191> <pack> INFO pool-2-thread-1 org.jboss.weld.environment.tomcat7.Tomcat7Container - Tomcat 7 detected, CDI injection will be available in Servlets and Filters. Injection into Listeners is not supported
Question 1: why is CDI injection not available in Listeners?
I looked into this answer, and it says Load on startup via @Startup. There is currently no equivalent to this in CDI.
Question 2: is the issue described in Question 1 a consequence of this?
Question 3: I am using org.jboss.weld.servlet:weld-servlet:1.2.0.Beta1.  Is there any update on startup support in later versions?
Related Questions I Looked startup class in Weld
Here is a workaround I discovered that can inject CDI beans when an application starts.
The requirement of the problem can be summarized as:
Solution outline line:
BeanManager.fireEvent(new SomeDummyEvent())
SomeDummyEvent and injects the CDI beanExample code:
@WebListener
public class ApplicationInitialisationListener implements ServletContextListener {
    private static final Logger LOG = Logger.getLogger(ApplicationInitialisationListener.class);
    @Override
    public void contextInitialized(ServletContextEvent event) {
        BeanManager beanManager = lookUpBeanManager();
        if (beanManager != null) {
            beanManager.fireEvent(new SomeDummyEvent());
            LOG.info("beanManager fired SomeDummyEvent.");
        } else {
            LOG.error("beanManager is null.  Cannot fire startup event.");
        }
    }
    @Override
    public void contextDestroyed(ServletContextEvent servletContextEvent) {
    }
    public BeanManager lookUpBeanManager() {
        try {
            // See reference below about how I came up with this
            InitialContext iniCtx = new InitialContext();
            BeanManager result = (BeanManager) iniCtx.lookup("java:comp/env/BeanManager");
            return result;
        } catch (NamingException e) {
            LOG.error("Could not construct BeanManager.", e);
            return null;
        }
    }
    public static class SomeDummyEvent implements Serializable {
    }
}
@ApplicationScoped
public class InitializationResourceBean {
    private static final Logger LOG = Logger.getLogger(InitializationResourceBean.class);
    @Inject
    HealthCheck healthCheck;
    public void listen(@Observes ApplicationInitialisationListener.SomeDummyEvent event) {
    }
    @PostConstruct
    public void init() {
        // Do something with healthCheck
    }
    @PreDestroy
    public void destroy() {
        // Do some other thing with healthCheck
    }
}
References:
http://struberg.wordpress.com/tag/cdi/
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