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.
 A: 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 $\log x > M$.
A: Hint:
show that if $x > \lceil\exp(M+1) \rceil$, then we must have
$$\ln(x)>M$$
A: Your error is that you have the wrong definition of limit.
Finite limit (with value $L$):

$\forall \epsilon > 0 :\; \cdots \implies |f(x) - L| < \epsilon$

Limit is $+\infty$

$\forall M:\; \cdots \implies f(x) > M$

Limit is $-\infty$

$\forall M:\;\cdots \implies f(x) < M$

You used the version for finite limits, rather than the version for a limit equal to $+\infty$.
A: One can use $\ln(x)=\int_1^x\frac{1}{t}dt\,$ (differentiable with $\ln'(x)=\frac{1}{x}$) and L'Hopital's (actually Bernoulli's) rule as follows:
Suppose that $\lim\limits_{x\rightarrow\infty}\ln(x)=L<\infty$ (as $\ln$ is a strictly increasing function, we only have two options: $\lim\limits_{x\rightarrow\infty}\ln(x)=\infty$ or $\lim\limits_{x\rightarrow\infty}\ln(x)=L<\infty$) and define $f(x)=L-\ln(x)$, which satisfies $\lim\limits_{x\rightarrow\infty}f(x)=0$. By L'Hopital's rule $$\lim\limits_{x\rightarrow\infty}xf(x)=\lim\limits_{x\rightarrow\infty}\frac{f(x)}{\frac{1}{x}}=\lim\limits_{x\rightarrow\infty}\frac{-\frac{1}{x}}{-\frac{1}{x^2}}=\lim\limits_{x\rightarrow\infty}x=\infty.$$
Hence, using L'Hopital's again $$0=\lim\limits_{x\rightarrow\infty}f(x)=\lim\limits_{x\rightarrow\infty}\frac{xf(x)}{x}=\lim\limits_{x\rightarrow\infty}\frac{f(x)-1}{1}=-1.$$ Therefore, $\lim\limits_{x\rightarrow\infty}\ln(x)=\infty.$
