It is established etiquette by the Joint Chiefs of Staff, United States Department of Defense, the U.S. Army, the U.S. Navy, and the U.S. Air Force that the closing “Respectfully yours” be reserved for the President of the United States (according to U.S. Army standards, this also extends to the first lady and the President_elect).
Anything that you'd use in an informal communication is inappropriate for a business letter. This includes slang, text_speak, emojis, and anything off_color or casual. If you're used to communicating mostly with friends, family, or even co_workers you've worked with for a long time, an appropriate closing for a business letter will probably feel pretty stilted at first. Don't worry about it – your colleague or business associate won't feel that way when he or she reads your correspondence. What seems unnatural to you will feel respectful and polite to the recipient.