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

*print*Print*list*Cite

Expert Answers

sciencesolve | Certified Educator

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.**