Homework SourseDetermine if f is differentiable at 0 a fx xsin1x if x does
Determine if f is differentiable at 0:
(a) f(x) =xsin(1/x) if x does not equal 0
0 if x = 0
(b) f(x) = x^3 sin(1/x) if x does not equal 0
0 if x = 0
Note: These are piecewise functions!
(a) f(x) =xsin(1/x) if x does not equal 0
0 if x = 0
(b) f(x) = x^3 sin(1/x) if x does not equal 0
0 if x = 0
Note: These are piecewise functions!
Solution
1) Clearly, F is continuous away from x = 0. As for x = 0, we need to check if lim(x?0) F(x) = F(0) <==> lim(x?0) x sin(1/x) = 0. This is indeed true by the Squeeze Law: Since 0 = |sin(1/x)| = 1 for all nonzero x, we have 0 = |x sin(1/x)| = |x| for all nonzero x. Since lim(x?0) 0 = 0 = lim(x?0) |x|, we can conclude that lim(x?0) |x sin(1/x)| = 0 ==> lim(x?0) x sin(1/x) = 0. Hence, F is continuous for all x.
