I frequently field requests from patrons at the Mesa FamilySearch Library looking for death certificates and obituaries. These documents are commonly sought for information on a death date of an ancestor. Most of these patrons requesting a death certificate are entirely unaware of the time periods during which these types of documents could be found. With respect to obituaries, they often begin and end their search with one local newspaper. In both cases, they are limiting their search, when other documents and records might provide the same, or even more information.
Commonly researchers are disappointed to learn that death certificates are a relatively recent innovation. In Arizona, for example, death certificates were not universally required by the State until after 1909 but there was not full compliance with the requirement until about 1926. Some death certificates date back before that time period, but there may not be one for any particular person.
Even in the Eastern states, implementation of uniform death registration occurred at relatively late dates. The state of New York, for example, began death records in 1847, but general statewide registration did not occur until 1880 and full compliance did not occur until 1890. So looking for a "death certificate" depends on the time period of implementation and the date of compliance by local record keepers. If you want to know the dates for other jurisdictions, I suggest looking in the FamilySearch Research Wiki under the topic of Vital Records for each jurisdiction. For example, search for Ohio Vital Records.
Obituaries are an entirely different issue. The presence or absence of an obituary in any given newspaper depends entirely on the circumstances of the person's death and the interest of the community and in some instances, if the family could afford to place the announcement. Commonly, patrons come asking for an obituary index. They believe that you can just look on a computer database and see a list of obituaries. Well, such indexes do exist in a spotty sort of way, but few of them go back more than the late 1970s or 1960s at the earliest and most of them only focus on one or two local newspapers. For each existing index, you must know which newspapers are included and time period for the coverage. Failing to find an obituary in an index is really no indication at all that such a document does not exist. In some instances obituaries may have been published in other county, or even regional or state newspapers.
Searching for obituaries should always include an many online digital newspaper collections as you can gain access to. For example, you should at least search the Newspaper Archive available from WorldVitalRecords.com and with a MyHeritage.com Record Search. This search is free in FamilySearch Centers. You can also search in NewsBank.com or its genealogy based program, GenealogyBank.com these resources may be available through a public library but are a subscription service. Also remember the Chronicling America Historic Newspapers on the Library of Congress website. There are also many state digital newspaper projects, notably, those in Utah, Colorado and other states. Do a general search for digital newspapers.
But don't stop there. Go to the Library of Congress and look at the US Newspaper Directory 1690 to the Present on the Chronicling America Historic Newspaper page. From there you can find a list of the newspapers published in the state and/or county during any time period. The list also has a link to all of the repositories that may have copies of the newspapers.
And there is still more. There are a lot of other types of sources that can give a death date. You may wish to look at the United States Vital Record page in the FamilySearch Research Wiki. You may also wish to look at United States Death Records in the Research Wiki.