# Math Help - Smooth function

1. ## Smooth function

Someone can help me attack this problem?

Let e: R into R by
e(x) = e^ (-1/x) if x > 0 and e(x) = 0 if x < or equal 0.
1) Show that e is smooth, that is, e has derivatives of all orders at all points x . Hint use L'Hospital and induction
2) Is e analytic?

2. No, it is not analytic. Because all derivatives are zero. Thus, it would mean f converges to the zero power series on any interval.