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Is (abs(x))^(2/3) differentiable at 0?
|x|^(2/3) --> When I approach with both the left and right hand limits for differentiation (that is, lim as x-->c of f(x) - f(c) / x-c, where c is 0 in this case), I get infinity for both limits. I know that in order for the derivative to exist at a point, the right and left hand derv. limits must be equal. But does this count? Can someone please assist me?
1 Answer
- Michael TLv 51 decade agoFavorite Answer
You could say that the curve does indeed have a vertical tangent at x=0, but we require the derivative be finite. We cannot speak of non-finite numbers being equal (well, at least not in calc 2). So, the answer is NO.
Part of the reason is that we want to use something called the implicit function theorem later on, and this requires the tangent line have a defined slope.
