Let f (x) be defined in a neighborhood of x = a, then a sufficient condition for f (x) to be differentiable at x = a is? A.lim (x approaches 0) [f (a + 2H) - f (a) Dlim (x tends to be 0) H [f (a + 1 / h) - f (a)] exists. Explain in detail why abd is not right (especially d)? A. LIM (H tends to 0) [f (a + 2H) - f (a + H)] / h exists, while LIM (H tends to 0) [f (a + H) - f (A-H)] / 2H exists C. LIM (H tends to 0) [f (a) - f (A-H)] / h exists. ABC option was forgotten just now, now add it. D is changed to dlim (H approaches infinity) H [f (a + 1 / h) - f (a)]

Let f (x) be defined in a neighborhood of x = a, then a sufficient condition for f (x) to be differentiable at x = a is? A.lim (x approaches 0) [f (a + 2H) - f (a) Dlim (x tends to be 0) H [f (a + 1 / h) - f (a)] exists. Explain in detail why abd is not right (especially d)? A. LIM (H tends to 0) [f (a + 2H) - f (a + H)] / h exists, while LIM (H tends to 0) [f (a + H) - f (A-H)] / 2H exists C. LIM (H tends to 0) [f (a) - f (A-H)] / h exists. ABC option was forgotten just now, now add it. D is changed to dlim (H approaches infinity) H [f (a + 1 / h) - f (a)]

There is a definition in a neighborhood of x = a, which means that the change of h will not be too big. So D is wrong (1 / h - > 0, H - > infinity, it's too wrong!). At the same time, x + H and X-H span x, which means that h is also relatively large, because if x + H is on one side of X, X-H should also be on the same side of X, so as to ensure that it is small enough. But x + H and X-H are on both sides, so they are wrong. The reason why a is wrong is that it does not describe the derivative of x = a, It's the derivative at x = a + h. even if h is small enough, it's still far from the real value of option C