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Would a native ever say "私は多忙{たぼう}です。" in a conversation? In what context? Is
it more of a word used in writing?

Isn't it a reasonable approximation of "私は中々忙しいです" (which is very
conversational)?
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2016-07-26T04:10:58.993
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2016-07-26T07:39:34.447
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question
word-choice
In what context does it sound natural to say / write "多忙"?
No, I don't think so. People would say 忙しい naturally.

Also, I noticed that if you are saying your own busyness, you won't use なかなか,
instead けっこう、とても、かなり. なかなか is more to 2nd/3rd person/ object, which is out of
your expectation. Although, grammatically and logically it is not wrong.
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2016-07-26T06:54:05.050
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2016-07-26T06:54:05.050
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Yes 多忙 is generally more formal than 忙しい, but you can safely use 多忙 in
everyday business conversations.

But note that 多忙 is not always semantically interchangeable with 忙しい. 多忙 means
having a full schedule over a relatively long period (at least for days,
usually for months or years). 私は多忙です doesn't mean "I'm busy (now)" but it
sounds more like "I am a busy businessperson." It may be acceptable in some
situations, but if you say this carelessly, it would give an arrogant
impression.

多忙 is mainly used (with some respect) referring to other person. If you want
to use 多忙 referring to yourself, you should be careful not to be rude. For
example, if you are a really busy person, you can say something like this:

> 申し訳ありませんが、多忙につき次回の会議は欠席いたします。
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2016-07-26T07:39:34.447
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2016-07-26T07:39:34.447
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answer