Question

Determine whether f'(0) exists.


***it's a peice-wise defined function***
f(x) =
x is not equal to 0....... xsin(1/x)
x is equal to 0......... 0

Answer 1

well for it to exist the left and right limits of xsin(1/x) would have to be 0, because otherwise the function would be nondifferentiable and would have no derivative at x = 0
You can basically do this by trying values and see if it approaches 0...
if x = 0.9 then y = 0.0175... x = 0.1 so y = 0.0173... it's safe to assume that the limit approaches 0 from the right
if you do the same on the left it works as well
therefore, f'(0) exists