Second Derivative - Limit

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

\sgn(x) = \begin{cases} -1 & \text{if } x < 0, \\ 0 & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}

The sign function is not continuous at null and therefore the second derivative for does not exist. But the above limit exists for :

\begin{align} \lim_{h \to 0} \frac{\sgn(0+h) - 2\sgn(0) + \sgn(0-h)}{h^2} &= \lim_{h \to 0} \frac{1 - 2\cdot 0 + (-1)}{h^2} \\ &= \lim_{h \to 0} \frac{0}{h^2} \\ &= 0 \end{align}

Read more about this topic:  Second Derivative

Famous quotes containing the word limit:

    “There is no limit to what a man can do so long as he does not care a straw who gets the credit for it.”
    —C.E. (Charles Edward)

    “We live in oppressive times. We have, as a nation, become our own thought police; but instead of calling the process by which we limit our expression of dissent and wonder “censorship,” we call it “concern for commercial viability.””
    —David Mamet (b. 1947)

    “Greatness collapses of itself: such limit the gods have set to the growth of prosperous states.”
    —Marcus Annaeus Lucan (39–65)