How many direct messages can I send out a day via broadcasts?

Twitter API limits will still apply:

Current Twitter limits

The current technical limits for accounts are:

  •  Direct Messages (daily): The limit is 1,000 messages sent per day.
  •  Tweets: 2,400 per day. The daily update limit is further broken down into smaller limits for semi-hourly intervals. Retweets are counted as Tweets.
  •  Changes to account email: 4 per hour.
  •  Following (daily): The technical follow limit is 1,000 per day. Please note that this is a technical account limit only, and there are additional rules prohibiting aggressive following behavior. Read about following limits and prohibited behavior
  •  Following (account-based): Once an account is following 5,000 other accounts, additional follow attempts are limited by account-specific ratios. 

These limits include actions from all devices, including web, mobile, phone, API, etc. API requests from all third-party applications are tracked against the hourly API limit. People who use multiple third-party applications with their account will therefore reach the API limit more quickly. 

These limits may be temporarily reduced during periods of heavy site usage. In such cases, we will post an update on the Twitter status site.