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Is x/x differentiable at x = 0

Now i know that it is not defined at x=0 but the function does approach the same limits from either side...From what i remember the limit does exist (what was the name of the rule that lets you do that)...but does that mean it is differentiable at x= 0 ???

Also can the same idea be extended to complex functions (say z/z)

Thanks in advance