Why is $\frac{\ln\infty}{\infty}$ equal to $\frac\infty\infty$? For me there is no intuitive explanation for this. Yes, i get that if you want to find the natural logarithm of a very high number (infinity) that the ln would be high too. But it does not grow nearly as fast as infinity itself, for example the natural logarithm of $\ln{100} \approx 4.60$ that number is 21 times as small as the number we are taking ln off. But anyhow, here comes the equation making my head jittery.

$$ \lim_{x\to\infty} \frac{\ln{\left(x\right)}}{x}=\frac{\infty}{\infty}$$

How, is this possible. Doesn't $\ln$ infinity grow at a much slower speed than infinity itself? 

I know how to do derivatives. I'm asking for an intuitive explanation. Of why we use that specific convention. 

In other words, why do we use an indeterminate form, and is there an intuitive explanation for why we might use infinity over infinity, or zero over zero?

Khans example. Showing that $\frac{ln\left(x\right)}{x} = \frac{\infty}{\infty}$
$\dfrac{\infty}{\infty}$ is a symbol that in this context just means the numerator and denominator both "approach infinity" (grow without bound). It's true that the numerator approaches infinity far slower than the denominator, which is why ${\displaystyle \lim_{x\to\infty}}\dfrac{\ln x}{x}=0$.

$\dfrac{\infty}{\infty}$ is called an "indeterminate form" because it doesn't tell you enough information to determine what the limit is.
If you use integration to find the average value of $y$ of the function $y = x^{-1}$ where $x$ is between $1$ and infinity, the result evaluates to $\frac{\ln(\infty)}{\infty - 1}$ and since the highest value of $y$ in this range is $\frac 1 1 = 1$, and the lowest value is $\frac 1 \infty = 0$, the value of $\frac {\ln(\infty)}{\infty - 1}$ lies between $0$ and $1$ (but not including $1$) so $\frac {\ln(\infty)}{\infty}$ is less than $1$.