if $f:[1, \infty)\to\mathbb{R}$, if the limit $\int^\infty _1 f(x)dx$ exists then $\lim\limits_{x\to \infty}f(x)=0$ [duplicate]

Question

I have this: if $f:[1, \infty)\to\mathbb{R}$ and the limit $\int^\infty _1 f(x)dx$ exists then $\lim\limits_{x\to \infty}f(x)=0$

How can I show this to be true, is it similar if it were $[0,\infty)$? Or am I incorrect in thinking this statement to be true?

Thanks to anyone, who can help!

Answer 1

A counter example: $$ \int_1^\infty \sin(x^2)dx $$ exists. However $$ \lim_{x\to\infty}\sin(x^2) $$ doesn’t exist.