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I want to ask a question and then answer it in my essay. How do I do this so that it is grammatically correct? Here is any example:

"Can you do X? Definitely. Can you do Y? Probably not."

The responses, "Definitely." and "Probably not." are not complete sentences. Are they appropriate to use in this context?
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I think the quotation marks in the original may have been there just to isolate the statement at issue.

If the dialogue represents a conversation, I like the one-word-sentence option.

"Can you do X?"

"Definitely."

"Can you do Y?"

"Probably not."

If it's an abstract or rhetorical dialogue which I guess it would be in an essay you might consider this (building on Mister Micawber's): Can one do X? --definitely. Can one do Y? --probably not. (Though I still, even here, personally like the one-word-sentence option.)
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Hi guys,

A well-placed rhetorical question can be a very effective device in writing. Howevr, if I may offer a personal thought on style, I'd recommend keeping such questions and conversational-style answers to a minimum in any essay that is supposed to be formal.

I think that, over-used, they can become irritating to the reader and can often lead to a less than clear expression of the writer's ideas.

Do you see what I mean? Sure, you do.

Clive
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Comments  
Your dialogue is out of context, but yes, you can probably use them. If the you is the writer, then I would use I:

"Can I do X?" -- definitely. "Can I do Y?" -- probably not.

That's how I would format it. There may be other options.
 davkett's reply was promoted to an answer.
 Clive's reply was promoted to an answer.
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I think yes, why not? It looks quite natural. It also depends upon context. Of course, you should develop this argument in your further writing to make it seem more logical. I also think that knowledge of these literary terms would also help you to write better.