Convergence of related improper integrals

Note that as $c \to \infty$,

$$\int_1^c \frac{f(x)}{x} \, dx + \int_1^c \frac{dx}{x f(x)} = \int_1^c\left(f(x) + \frac{1}{f(x)} \right) \frac{dx}{x} \geqslant 2 \int_1^c\frac{dx}{x} \to +\infty$$

Both integrals on the LHS cannot converge.

Tags:

Integration

Improper Integrals

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