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Should you ever use its without the apostrphe (it's)?
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It's possible but it has a different meaning.

It's = IT IS (a contraction)
Its = a possessive adjective.....used to show possession.

My dog is black. It's very big and eats cats.
here "It's" is a shortened form of IT IS

My dog eats its food very quickly.
here "its" refers to the fact the food belongs to the dog.

It's eating its food quickly.
Comments  
You might want to check this thread: Apostrophes
 Woodward's reply was promoted to an answer.