Find functions $f(n)$ and $g(n)$ such that $f(n)\in\Omega(g(n))\setminus\Theta(g(n))$, but $\lim_{n\to\infty}$ does not exist.

Question

I found an answer but I'm not sure if it is correct:

Let $f(n)=(2+\sin n)\cdot n$ and $g(n)=\ln n$. $0\leq g(n)\leq f(n)$ $\forall n\gt0$ so $f(n)=\Omega(g(n))$. But $f(n)\neq\Theta(g(n))$ because $f(n)$ will always grow faster than $g(n)$ regardless of a constant multiple. Additionally $f(n)\neq\omega(g(n))$ because $\lim_{n\to\infty}\frac{(2+\sin n)\cdot n}{\ln n}$ diverges and $7\ln4\gt(2+\sin4)\cdot4$.

Would these functions meet the aforementioned requirements?