How to solve this example?


Sep 2016
This seems a bit like homework, so you should show what you have done already and just prove you have thought about the problem some. It's a cute problem though, so here is a hint.

Hint: Can you see that $\lim\limits_{t \to \infty} \left( f(t) - t \right) = 0$? If not, spend some time working out why this is true. This gives the intuitive guess by taking derivatives on both sides that we should have
\[ \lim\limits_{t \to \infty} f'(t) = 1 \]
Of course this guess is not necessarily true since it's often not allowed to exchange limits like I have done. However, this should give you the idea to check whether or not this limit exchange is justified. In this case, it is. This is, in fact, a consequence of the first limit. To see this, let $g$ be the integrand. Then you can easily obtain the result
\frac{d}{dt} \int_t^{f(t)} g(x) \ dx = g \circ f(t) \cdot f'(t) - g(t) = 0.
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