Let's assume that $f:[1,\infty) \to [0,\infty)$.
Can someone provide an example of a function where the improper integral $\int\limits_1^{\infty} f(x)dx$ doesn't exist because we can find two sequences $\left(a_n\right)_{n\in\mathbb {N}}$ and $\left(b_n\right)_{n\in\mathbb {N}}$ with $\lim\limits_{n\to\infty}a_n=\lim\limits_{n\to\infty}b_n=\infty$ such that $\lim\limits_{n\to\infty}\int\limits_1^{a_n} f(x)dx<\infty$, $\lim\limits_{n\to\infty}\int\limits_1^{b_n} f(x)dx<\infty$ and $\lim\limits_{n\to\infty}\int\limits_1^{a_n} f(x)dx \neq \lim\limits_{n\to\infty}\int\limits_1^{b_n} f(x)dx $?
Best Answer
I don't think such a case is possible, indeed $$g : x \rightarrow \int_1^x f(t)dt$$ is a monotonous function and as such admits a limit as $x \to \infty$. This limit can be either finite or $+\infty$ but in either case for any sequence $x_n$ that goes to $+\infty$, we have that $lim_{n\to\infty}g(x_n) \to lim_{x\to\infty}g(x)$.