6.5. Differentiable Functions
Use the Mean Value theorem to show thatDefine the function f(x) = . Then f is continuous on [0, ) and differentiable on (0, ). By the Mean Value theorem there exists a number c such that
= f'(c)for c between x and x + 1. But then
0 < - = 1/2 * 1 /As x goes to infinity, so does c (it is always bigger than x). The left side of this equation goes to 0 as c goes to infinity. Therefore, the right side must also go to zero.