# 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.

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?

• Your limit here is infinity, is that really what you meant? Oct 7, 2019 at 19:05
• No, I was looking for a limit that does not exist (even in the sense of being infinite) such as $f(n)=(1+\sin n)\cdot n$ and $g(n)=n$, $\lim_{n\to\infty}\frac{f(n)}{g(n)}=\lim_{n\to\infty}(1+\sin n)$ whose limit does not exist. Oct 7, 2019 at 19:09
• Then your example does not work, since $g=o(f)$. A suggestion: you could make $g(n)=\sqrt{n} + (1-(-1)^n)n$. Oct 7, 2019 at 21:41