Suppose that I want to show that $g(n) \in O(f(n))$ as $n\rightarrow\infty$ for functions $f(n),g(n)>0$. I know that this means that there exists a constant $c$ such that for all sufficiently large $n$ it holds that $g(n) \leq c f(n)$. Moreover, according to wikipedia this is equivalent to $\lim\sup_{n\rightarrow\infty}\left|\frac{g(n)}{f(n)}\right| < \infty$.
However, even if I can show that $\lim_{n\rightarrow\infty}\left|\frac{g(n)}{f(n)}\right| = c < \infty$, this also implies $g(n) \in O(f(n))$, since from this it follows that $g(n) < (c+\epsilon)f(n)$, for sufficiently large $n$ and any $\epsilon>0$.
Could someone give me an example where the first condition (using $\lim\sup$) holds but the second one (using only $\lim$) does not?