HypeAuditor doesn't limit you to how many bulk emails you can send. However, email providers have limits built on optimal security for you and your recipients.
Gmail limits
Google (Gmail) lets you send emails to a maximum of 500 recipients daily through Gmail.
According to Google's FAQ, you may receive a "You have reached a limit for sending mail" email if you exceed the limit. When you get this error, you should be able to send emails again within 24 hours.
Outlook limits
300 emails is the limit for a verified Outlook.com email address with a good reputation.
If you exceed this limit, Microsoft may temporarily restrict sending from your account. It is recommended to wait 24 hours before attempting to send again.
To avoid hitting provider limits, consider spreading your bulk sends across multiple days when reaching out to a large number of influencers.
If you haven't connected your mail agent yet, see Why I can't connect my mail agent to Influencer Outreach? for setup guidance.