ことなしに is a formal expression meaning "without doing ~". It is used to indicate that something happens or is done without a particular action taking place. It attaches to the dictionary form of a verb.
This pattern is more literary and formal than the more common 〜~ないで or 〜~ずに, and it tends to appear in written language, speeches, and formal contexts. It often carries a nuance of criticism or surprise – implying that the omitted action should have been done, or that it is noteworthy that it was not done.
All three expressions mean "without doing ~", but they differ in register and nuance:
A closely related form is ことなしには, which adds the particle は to create a conditional or contrastive nuance, meaning "without doing ~, one cannot…" or "unless one does ~".