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I have been wondering whether the following limit is being used somehow, as a variation of the derivative:

$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$

Edit: I know that this limit is defined in some places where the derivative is not defined, but it gives us some useful information.

The question is not whether this limit is similar to the derivative, but whether it is useful somehow.

Thanks.

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    I think it is used in numerical applications to linearize problems involving the derivative of a function. The trouble is that finding the derivative as a function of $x$ is really hard in general. So instead we can discretize the problem and approximate the derivative by this expression for very small $h$. (At least I have seen this in my introductory course on numerical methods)2011-09-18
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    @Sam, As far as approximation and discretization are concerned, it seems exactly as convenient to use $n(f(x+1/n)-f(x))$ than $\frac12n(f(x+1/n)-f(x-1/n))$.2011-09-18
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    @Didier: Not necessarily. For example when we want to discretize the first order ODE $\dot x = x$, we will get a skewsymmetric matrix with the second approach, which may or may not have advantages over the less symmetric first variant (but I definitely do not know, so it's just a feeling)? Anyways: I have seen the expression the OP is asking about being used to discretize an ODE.2011-09-18

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