Why does x approach 0 faster than ln(x) approaches infinity?

[tmt1]

I have

$\lim_{{x}\to{0}} lnx \cdot x$

$x$ approaches 0, and $lnx$ approaches $\infty$.

How can I reason about this.

I suppose $x$ approaches 0 more quickly than $lnx$ approaches $\infty$ , therefore it is zero. Is this accurate? How can I prove this.

 

[MarkFL]

I would write it as:

$$L=\lim_{x\to0}\left(\frac{\ln(x)}{\dfrac{1}{x}}\right)$$