How can I use f'(x)= lim x->0  [f(x+h)-f(x)]/h to determine whether f(x)=x |x| is differentitable at x=O?

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beckden | High School Teacher | (Level 1) Educator

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f '(x) = lim(h->0, ((x+h)|x + h| - x |x|)/h) at x = 0 this reduces to

f '(0) = lim(h->0 (h|h| - 0)/h) = lim(h->0 |h|)

This limit is zero both on the - side and + side, so the limit exists and therefor the derivative exists and it is zero.

This is not true of f(x) = |x| because if you do the same calculation you will get - limit is -1 and the + limit is +1 and since the derivative is discontinuous at x = 0 the derivative does not exist.

To repeat, f(x) = x|x| the limit is zero both on the - side and + side, so the limit exists and therefor the derivative exists and it is zero at x = 0.

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