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_Onegaishimasu_ is used by Japanese people a lot. What does it mean, in what
context is it used, and how do I respond to this?

I found on the web that it means:

> Please take care of me.
>
> I am in your care.
>
> Do me this favor.
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2016-10-13T10:35:14.467
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2016-10-14T07:13:12.823
2020-06-17T08:18:27.500
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meaning parts-of-speech
Onegaishimasu meanings, reply and when to use
If you translate literally, onegai means request, shimasu means to do(polite).
So it's basically a polite way to make a request.  
As you said, it is used a lot and in many context.

I could make a list of way to use it, but it would be quite a long one.

those 2 might be more adequate for the expression "yoroshiku onegaishimasu".

> Please take care of me.  
>  I am in your care.
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2016-10-13T10:40:34.353
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You can think of it as the word ¨please¨ in English. If someone says おねがいしますね
to you, they are always asking you to do something or expect something from
you.

For example if a friend tells me that she likes chocolates and I say, I will
buy for you. Then she might say, お願いします。 meaning, ¨please do¨, or ¨that would
be great¨.

If you want to imply that the other person should actually do something
because it's important, you can use おねがいしますよ which is a bit stronger.  
For example I can ask my brother to pick up my kids from school tomorrow and I
can at the end of the sentence say おねがいしますよ as a way to emphasise that he
shouldn't forget this. I think this is the example of what you posted in your
question ¨Please take care of me.¨

You can respond to the おねがいします as どうぞ ¨here you go¨ when they are asking you
for something, or you can answer はい、わかりました ”understood” when it's an important
request that you will do later.

Hope this helps.
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2016-10-13T21:26:55.363
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2016-10-14T07:05:59.443
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answer