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Q. $ \lim_{x \to k}x- [x] $is where k is an integer, is equal to

Solution:

$ \lim_{x \to k} (x- [x]) $ does not exist
$ \lim_{x \to k-}x- [x] = k -[k-] =k - (k-1) = 1$
$ \lim_{x \to k+}x- [x] = k -[k+] = k- k =0$