Originally Posted by

**Jose27** No, the correct one is the limit I wrote. What I'm trying to do is show that $\displaystyle x^c$ grows at a faster rate than $\displaystyle ln(x)$, which I did, because the limit is $\displaystyle \infty$. the limit you wrote indeed says nothing about what you're trying to do, but what you want is to compare $\displaystyle x^c$ and $\displaystyle ln(x)$.

Now if you want a proof based on the inequalities you gave, then, I'm not sure your inequalities are sharp enough, since the best you can get from them is $\displaystyle \frac{x}{c!} \leq ln(x)$