Which is correct if I want to request for a pen?
- Can I have your pen please?
- May I have your pen please?
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Can primarily expresses possibility and ability and, secondarily, permission. May expresses primarily possibility and, secondarily, permision and volition. In seeking permission, as in your examples, the use of may is much more formal and polite than can and is used rather less.
However, both 'Can I have your pen please?' and 'May I have your pen please?' are blunt ways of making a request. In practice, a native speaker, at least of British English, is much more likely to say something like 'You don't happen to have a pen I could borrow, do you?'
NOAD has a usage note that reads:
Is there any difference between can and may when used to request or express permission, as in : may I ask you a few questions? or : can I ask you a few questions? It is still widely held that using can for permission is somehow incorrect and that it should be reserved for expressions denoting capability, as in : can you swim? Although the use of the 'permission' sense of can is not regarded as incorrect in standard English, there is a clear difference in formality between the two verbs: may is, generally speaking, a more polite way of asking for something and is the better choice in more formal contexts.
I suspect many other English dictionaries have similar notes under their entries for may or can.
Why not ask etymology for some help here?
The word can comes from Proto-Germanic kunnan
to be mentally able, to have learnt
Can is about ability, skills, knowing how to do things, whereas the word may comes from Proto-Germanic root mag-, infinitive maganan, from Proto-Indo-European magh-
to be able, have power
May is about power, the position a person is in to grant or refuse another person permission to do something, says the Online Etymology dictionary.
Can I go to the bathroom?
– etymologically – is not really appropriate for asking permission as it means something like Do you think I know how to go to the bathroom?, which might sound rude, even! On the other hand,
May I go to the bathroom?
– etymologically again – means something like Would your highness allow me to go to the bathroom? and, in the classless world we live in (?) we are not really ready to your-highness anyone if we can avoid it! Well, little children cannot really avoid it for a question of stature and status, but, when you have grown up, using may is felt as humiliating, as if you were kneeling in front of someone.
Could I go to the bathroom?
represents a middle ground between rudeness and self-humiliation, which can satisfy both the speaker and the listener!
Using the modal auxiliary can in the conditional, 'unrealising' the present, making it hypothetical, dampens the shock. I am asking about objective conditions (do you think I know how to) but using the conditional instead of the present puts more emphasis on the listener, on their ability to assess the situation, which is a tacit acknowledgement of their authority.
The word ‘may’ is used to convey the following sense:
The word ‘may’ shows possibility. It means the happening of something is not sure. There is a confusion whether it will happen or not.
It may rain today.
He may go to market in the evening.
She may meet you in the afternoon.
It may happen.
The wind may blow violently.
The word ‘may’ is used to show a requesting manner.
May I come in, Sir?
May I take your pen?
May you leave me for one hour?
May I help you?
This word is also used to order someone. Examples:
You may go now.
You may do this thing.
You may pluck a flower.
You may take my pen.
Finally as per your question
'Can I have your pen, please?' is a wrong sentence as there is nothing about ability.
'May I have your pen, please?' is the right one.