Interactive Real Analysis
- part of Examples 6.5.10(c):
Use the Mean Value theorem to show that 
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Define the function f(x) =
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.
Then f is continuous on [0, 
)
and differentiable on (0, ).
By the Mean Value theorem there exists a number c such that
for c between x and x + 1. But then= f'(c)
0 <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.-
