I face with this sentence in dictionary:
He didn't look like the sort of man you should entrust your luggage to.
Is it possible to eliminate a preposition being at the end of sentence without hurting the meaning? and why we use it?
... man to whom you should entrust your luggage.
The meaning is the same, but the style is now "19th century schoolbook" English.
It has been discussed before, but the "rule" about not ending a sentence with a preposition was made to emulate Latin grammar rules in English. There is no such rule in native English. It is (as Churchill is reputed to have quipped) "...pedantry up with which I will not put."