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Show that f haven't primitives. f=0, x=<0 f=sin(1/x)-1/xcos(1/x), x>0f=0,...

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mouhham | Student, Undergraduate | (Level 1) eNoter

Posted July 15, 2012 at 12:44 PM via web

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Show that f haven't primitives.

f=0, x=<0

f=sin(1/x)-1/xcos(1/x), x>0

f=0, x=<0

f=sin(1/x)-1/xcos(1/x), x>0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted October 19, 2012 at 2:30 PM (Answer #1)

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You need to test the continuity of the given function at x = 0 such that:

`lim_(x->0, x< 0) (sin(1/x)-(1/x)cos(1/x)) = oo`

`lim_(x->0, x>0) 0= 0`

`f(0) = 0`

Since `lim_(x->0, x< 0) (sin(1/x)-(1/x)cos(1/x)) = oo != 0 = lim_(x->0, x>0) 0` , then the function is not continuous at x=0, hence, it is not differentiable at x=0 and it has no primitives at x = 0.

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