Example 2

We have $\left( \frac{e^t}{x} \right)' = \frac{e^t}{x} \left( 1 - \frac{1}{x} \right) \sim e^t / x$; since $e^t / x$ tends to $+\infty$ with $x$, one deduces from prop. 6 of V, p. 230, that $\int_{1}^{x} \frac{e^t}{t} \, dt \sim e^x / x$.