Use Taylor's theorem to show a function is differentiable at x=0

[tobinator250]

Homework Statement

Use Taylor's theorem to estimate |(e^x)-x-1| for 0≤x≤1. Thus prove that if a>(1/2) then:

f(x)=(1-|x|^a)*(e^x)^a is differentiable at x=0

Homework Equations

The Attempt at a Solution

So |(e^x)-x-1|=(x^2)/2+(x^3)/6+(x^4)/24...

But I don't see how this helps, I have considered using the Lagrange remainder as well but again I can't see how that would help either. Any help would be greatly appreciated.

 

[HallsofIvy]

Since the lowest power of x in that formula is 2 so, for x less than 1, it is less than x.