Prove that $\lim_{x\to\infty} (\ln x) = \infty$ Can someone help me prove that the function $\ln(x)$ diverges to infinity as $x$ approaches infinity. I tried using the definition to show that $\lvert \ln(x) -∞ \rvert < \epsilon $ where $\epsilon > 0$ and there exists a number $N$ which is in the set of natural numbers where $x > N$, proving that it does indeed diverge to infinity but I can't get any further.
Any help is appreaciated.

I guess you want an analysis using the definition of divergence at infinity. 
Let $M > 0$; and note that $\log x > M$ if $x > e^{M}$. 
This shows that for every $M > 0$ there is some $X > 0$ (take $X := e^{M}$, say) such that $x > X$ implies $\log x > M$.
