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> Question: Mention (True or False) Let $f:[1,\infty)\to\mathbb{R}$ and $f(x)=x\int\_1^{x}\frac{e^t}{t}dt-e^x$ then f(x) is an increasing function and $\lim\_{x\to \infty}f(x)\to\infty.$
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Differentiating $f(x)$ we can conclude that $f(x)$ is increasing. Using $e^t=1+t+\frac{t^2}{2!}+\dots$
Then $$f(x)=x\int\_1^x(\frac{1}{t}+1+\frac{t}{2!}+\frac{t^2}{3!}+\dots)dt-e^x=x(\ln(t)+t+\frac{t^2}{2(2!)}+\dots)\_1^x-e^x=x(\ln(x)+x+\frac{x^2}{2(2!)}+\dots-1-\frac{1}{2(2!)}-\dots)-e^x\\$$

It seems that for $x\to\infty$ we will have $f(x)\to-\infty$ because of the rate of growth of the exponent .

The argument that I have mentioned is merely an intuition, this does not seems correct . Thanks.