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What "limit exists"? The limit you look at to determine if f(x) is continuous at x= a, is $\lim_{x\to a} f(x)$ while the limit you look at to determine if f(x) is differentiable at x= a is $\lim_{h\to a} (f(a+h)- f(a))/h$. It is easy to show that if a function is differentiable at x= a, it must be continuous but the other way is not true.