we learned about limit at infinity, after we studied about supremum and infimum last month, and I thought about something I'm struggling to think whether it's true or not.
If I know that a function has a supremum, is it always true that $\lim \limits_{x \to \infty}f(x)=sup(f(x))$?
Say$\ f(x)<1$ for every $\ x\geq$ 0, and I am given that$\ sup(f(x))=1$, does it mean that $\lim \limits_{x \to \infty}f(x)=1$?
No. Take, for instance$$\begin{array}{rccc}f\colon&[0,\infty)&\longrightarrow&\mathbb R\\&x&\mapsto&\dfrac{x\sin x}{x+1}.\end{array}$$Then $\sup\bigl(f(x)\bigr)=1$, but the limit $\lim_{x\to\infty}f(x)$ doesn't exist.