Tasks that are configured to email or SMS automatically will never appear in the task list because they are completed the moment that their communication action is completed. In order to verify that a task has generated and a communication has been sent for a member, head to that member's profile. Then look under their Communication tab.
In this example, you can see that this facility has an automated task set up to send a Welcome! email to member's whenever they start a new membership. First the task is generated, and then the email is sent.
If you can see the task in the list, but there is no corresponding email or SMS, ensure that the Task Type has been configured to send automatically. Settings > Task Setup >  > Auto Email / Auto SMS