## 6.5. Differentiable Functions

### Examples 6.5.10(c):

Define 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

for= f'(c)

*c*between

*x*and

*x + 1*. But then

As0 < - = 1/2 * 1 /

*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.