Exercise 3, § 3

Functions of a Real Variable, V, 260

Give an example of two functions which are positive, decreasing, tend to 0 as $x$ tends to $+\infty$ , comparable to order 1 and such that $xf(x)$ and $xg(x)$ are not comparable to order 1 (take $f$ and $g$ equivalent and such that

\frac{f'(x)}{f(x)} + \frac{1}{x} \quad \text{and} \quad \frac{g'(x)}{g(x)} + \frac{1}{x}

are not comparable).

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