There is a very good chance that it's written in to your lease that 30 days notice must be given. Not only could the 30 days be considered a courtesy to the landlord, but it also gives them the opportunity to begin advertising the vacancy so they can rent your current residence with the smallest vacancy gap possible. Most apartment complexes, or landlords, normally send out a reminder mailer about 60 days out to let you know as well. I realize that your landlord, may not have done this.
Regardless, your best course of action is to review the lease agreement that you signed when you moved in. If you don't still have a copy, your landlord should have one that can be copied and given to you. The lease will dictate whether 30 days notice is required at the end of the lease term, or only if you were to cancel your lease early.
I require all my tenants to give a 60 day notice because at that point a tenant knows what they are really intending to do. So it helps us both out.
If you want to better understand your rights, google landlord/tenant law colorado. It should bring you to some valuable resources.
A 30 day notice is standard. Again it protects you as well as the landlord. You can't be charged with an extra month, and he gets to advertise and prepare for a new renter to come in on October 1st. So it is normal, standard and fair. So yes you do need to give a formal 30 day notice in writing.
Robert McGuire ASR
Your Castle Real Estate
Search Denver Homes
Direct - 303-669-1246
Do not forget to do your walk-through, take photos, and have your landlord sign off that you are leaving the property in acceptable condition!
Like the others have stated, read your lease, it will answe 99% of most tennents questions.
If you are moving in 30 days, be sure to get the notice in writing to the landlord.