You're welcome, Emily.
My suggestions are underlined.
Quote:
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As the meeting progressed, I met with more problems. (Correct me, whenever I make a mistake during my retelling of the event.) The meeting had been going on for some time before I entered, and I wanted to know if I could interrupt. What should I have asked? Please note, it was a very formal business meeting, so my question should be an appropriate one.
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What about?
- Excuse me, my apologies for interrupting / May I interrupt?
Quote:
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After that, I introduced my company to my client, and found again it was difficult to start the introduction. I had in mind the words “May I introduce my company to you first?”, or “I’d like to introduce my company to you first.” Are those words appropriate?
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Well, "May" is a formal request, and what you're looking for is something more business-like, so why not try?
- First, allow me to introduce you to (the members of) my company.
In short,
Excuse me, may I interrupt? I'd like to introduce / Allow me to introduce