TITLE: find the limit of this problem
QUESTION [5 upvotes]: Prove that $\int\limits_1^x e^t/t dt \sim (e^x)/x$ as $x \longrightarrow \infty$. any ideas of how to approach this problem. I can not seem to evaluate the integral either so i am real stuck in this problem

REPLY [4 votes]: By integrating by parts we get
$$\int_1^x\frac{e^t}{t}dt=\frac{e^t}{t}\Bigg|_1^x+\int_1^x\frac{e^t}{t^2}dt$$
and using a second integration by parts we get
$$\int_1^x\frac{e^t}{t^2}dt=\frac{e^t}{t^2}\Bigg|_1^x+2\int_1^x\frac{e^t}{t^3}dt$$
and we have
$$\int_1^x\frac{e^t}{t^3}dt\le e^x\int_1^x\frac{dt}{t^3}=\mathcal O\left(\frac{e^x}{x^2}\right)$$
hence we conclude
$$\int_1^x\frac{e^t}{t}dt=\frac{e^x}{x}+\mathcal O\left(\frac{e^x}{x^2}\right)$$