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Thread moved from the technical forums to the schoolwork forums

**Summary::**x

**Problem:**

Let ##f:[0, \infty) \rightarrow \mathbb{R}## be a positive function s.t. for all ## M > 0 ## it occurs that ## f ## is integrable on ## [0,M] ##. Which of the following statements are true?

A. If ##\lim _{x \rightarrow+\infty} f(x)=0## then ##\int_{0}^{\infty} f(x) d x## exists and is finite.

B. If ##\int_{0}^{\infty} f(x) d x## exists and is finite then ##\lim _{x \rightarrow+\infty} f(x)=0##.

C. If ##\int_{0}^{\infty} f(x) d x## exists and is finite then ##\int_{0}^{\infty} f\left(x^{2}\right) d x## exists and is finite.

D. None of the above.

**Context:**I'm getting ready for an exam and I'm solving past exams. The above question was part of a past-exam, but I don't have any answers so I'm asking here to be , can you please help? I'm not sure.

**Attempt:**

I marked 'D' as the answer, here are my counter examples for A,B,C:

Counter-example for ## A##:

## f(x)=\begin{cases} 0 &\text{if}\; x \in [ 0,1] \\\\ \frac{1}{\sqrt{x}} & 1<x \; \end{cases} ##

Counter-example for ## B##: ## f(x) = \arctan x ##

Counter example for ## C##: the same ## f ## as in the counter example for A.