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One possible method is to use the following command:

tail -f myfile.txt | gzip > myfile.txt.gz

This command will continuously monitor the file myfile.txt for new lines using tail -f and then pipe the output to gzip, which compresses the data and sends it to the file myfile.txt.gz. The > operator redirects the compressed output to the file.

Note: If you want to remove the uncompressed file after the compression, you can use rm myfile.txt after the above command. However, be careful when removing files, as it may result in data loss or unexpected consequences.