Integration of exponential function having limits upto infinity

Question

I am just stuck with this problem: $$ \int_{0}^{\infty}{\mathrm{e}^{x/a} - \mathrm{e}^{b/x} \over x}\,\mathrm{d}x $$ It is also given that $ab$ is not equal to $0$.

Answer 1

See that

$$x\to\infty\implies\frac{e^{x/a}-e^{b/x}}x\to\infty$$

So that the integral cannot converge.

Similarly,

$$x\to0^+\implies\frac{e^{x/a}-e^{b/x}}x\to-\infty$$

In other words, theres all sorts of things preventing this integral from being evaluable.