Your gown shall be unpicked, but I do not remember its' being settled so before.
—Jane Austen, letter to Cassandra Austen, 7 October 1808
Many struggle with it’s vs. its (and some of us even find occasion to wrangle in an occasional its’). If you are one such person, take comfort in the fact that Jane Austen was likewise befuddled—although she at least had the excuse that the rules governing this matter were not yet entirely set.
A quick refresher: it’s is used when the writer intends to provide a shortened form of "it is" ("it’s a shame"); its is used when indicating possession ("the museum is failing, as its membership is declining"); its’ is generally avoided.
A look at Austen’s correspondence shows that she paid very little attention to the difference between these forms of the word. In an 1814 letter to her older sister, Cassandra, Austen wrote "It may take it’s chance"; in an 1808 letter, she used the correct "Our evening was equally enjoyable in its way."
No matter which version of this mistake one makes, there will be at least one authority from the past several hundred years of English who would have found it correct. The exception is James Buchanan, the author of A Regular English Syntax, who in 1767 declared "It’s for it is is vulgar; ‘tis is used."