You should write an apology letter when you have inconvenienced or offended someone. Humbling yourself is rarely pleasant, but when the occasion calls for it, you should not hesitate to write an apology letter. This sample letter can make it easier; it demonstrates how to write such a letter while retaining a relationship with the inconvenienced person.
114 Church Street
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I am extremely sorry that I forgot about our lunch date. I was so busy at work that day that it completely slipped my mind. To make amends, I would love to treat you to lunch next Wednesday at that new Italian restaurant, Julie's. Does 12:30 p.m. work for you? I have marked this date in my planner so that I will be sure to see it. Again, I apologize for missing our appointment.