I was thinking about the following problem:
Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists. Then which of the following statements are correct?
(a) If $\lim_{x\to\infty}f(x)$ exists, then $\lim_{x\to\infty}f(x)=0,$
(b) The limit $\lim_{x\to\infty}f(x)$ must exist and is zero,
(c) In case $f$ is a nonnegative function, the limit $\lim_{x\to\infty}f(x)$ must exist and is zero,
(d) In case $f$ is a differentiable function, the limit $\lim_{x\to\infty}f'(x)$ must exist and is zero.
If I take $f(x)=e^{-x}$, so that the given condition is satisfied then we see that options (a) and (c) are correct. But I am not sure about the choice given in (b) and (d). But If I have to prove it in general, then how can I prove it? I mean an alternative better approach. Please help. Thanks in advance for your time.
Best Answer
Statement (a):
Let's say $\lim_{x\to \infty} = a$. if $a\neq 0$, then by definition of limit, there is a number $X$ such that for any $x \geq X$, we have $|f(x)| \geq a/2$. This means that from that point on, the integral $\int_0^xf(x)\,dx$ will increase (or decrease, depending on whether $a$ is positive or negative) by at least $a/2$ for each unit we increase $x$, and thus the improper integral doesn't exist. So if $a$ exists and the integral is finite, then $a = 0$ by this contradiction.
Statement (b) and (c):
I do these together because they can be disproven with the same counter-example. Start out with the function that is constantly $0$. Now, take the interval $[0.5, 1.5]$, and raise the graph as a triangle with a top in the point $(1, 2)$. Now the total integral is $1$. Then for the interval $[1.75, 2.25]$, raise a triangle so the top is at $(2, 2)$. The total integral is now $1.5$. Continue doing this around every integer point along the $x$ axis, always halving the interval length. You will end up with a function that integrates to $2$, but has bumps all over with height $2$. So no limit. It is also non-negative, so it fulfils (c).
Statement (d):
Taking the function above, you can "smooth out" the corners so that the function is differentiable, and the change to the total integral is finite. Now you have a differentiable function which has derivatives of whatever size you want, if you just go far enough out, so the derivative has no limit.