Solving a Math Problem: Demystifying the Unknown

[glmrkl]

Homework Statement

http://puu.sh/cKVxE/fb13f83a75.png
In this image, I have no idea what the math behind this problem is. What exactly is happening here?

Homework Equations

The Attempt at a Solution

I multiplied n/10 in order to get 10000n^9 as a denominator but the ln(2) confuses me; therefore, I cannot get to the third part of the problem.

 

[RUber]

It appears that in each step they are taking the derivative of the [edit] top and bottom with respect to n (L'Hopital's Rule).
Taking 10 derivatives leads to the last limit, which is clearly unbounded.

 

[RUber]

If you have not seen the exponential derivative before, it just requires a few steps:
$\frac{d}{dn} 2^n =\frac{d}{dn} e^{\ln 2^n}=\frac{d}{dn} e^{n\ln 2}=e^{n\ln 2}\ln 2=2^n \ln 2$

 

[glmrkl]

Thanks a lot!