According to the documentation there is a limit of 1000 emails per day if sent from APEX or API.
If I had configured an email relay, would sending out an email from APEX count towards the email limit again?
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Activating email relaying will not remove the limits on mass email.
The reason for this is that the salesforce.com email servers still process the email and thus the bulk load is still on SFDC side. Once processed the email is then passed (or relayed) to your email server before sending.
These limits are enforced so that the load does not become too large for sfdc servers to handle and cause performance issues for other customers
Source : here
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