How to check instantly the changes made by user in the Google Docs?

Unfortunately there are no onEdit(), onChange() triggers for Google Docs.

The triggers available for Google Docs are the following:

• onOpen() triggers both simple and installable;

• time driven triggers;

• onInstall() simple triggers.

What you can do instead is file a Feature Request on Issue Tracker where you specify the details needed.

Reference

• Apps Script Trigger;