If limit exists, how to judge LIM (△ x → 0) [f (x0 + △ x) - f (x0 - △ x)] / △ x = f '(x0) error Wrong, it should be LIM (△ x → 0) [f (x0 + △ x) - f (x0 - △ x)] / 2 △ x = f '(x0) | Math enthusiast | Mathematical art

If limit exists, how to judge LIM (△ x → 0) [f (x0 + △ x) - f (x0 - △ x)] / △ x = f '(x0) error Wrong, it should be LIM (△ x → 0) [f (x0 + △ x) - f (x0 - △ x)] / 2 △ x = f '(x0)

Because f '(x0) means that f (x) is differentiable at x0, from which we can see that the function f (x) must be defined at x0
However, only LIM (△ x → 0) [f (x0 + △ x) - f (x0 - △ x)] / 2 △ x is known
It does not indicate whether f (x) is defined at x0, so it is wrong
Definition of Derivative
F '(x0) = Lim [f (x) - f (x0)] / (x-x0). The limit process is x → x0, which shows that f (x) is defined in x0!

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