Limit
It is possible to write a single limit for the second derivative:
The expression on the right can be written as a difference quotient of difference quotients:
This limit can be viewed as a continuous version of the second difference for sequences.
Please note that the existence of the above limit does not mean that the function has a second derivative. The limit above just give a possibility for calculating the second derivative but does not provide a definition. As a counterexample look on the sign function which is defined through
The sign function is not continuous at null and therefore the second derivative for does not exist. But the above limit exists for :
Read more about this topic: Second Derivative
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