You’ve requested the example.txt file. That file doesn’t exist/exists.
Which one should I use? Why?
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The right form is: The file doeSn't exist.
Gramatical explanation: always when you're forming a positive sentence about a 3rd person (singular only) in present simple tense, you have to add the letter s to the verb which describeS what he doeS.
Butwhen you're forming a negative sentence, as in your example, then this letter S goeS to the negative auxiliary verb instead to the verb itself. That's why in the sentence "The file doesn't exist" we don't add S to the verb exist, since this leeter already in the negative auxiliary verb (doeSn't rather than don't).