I read the following and I am not sure I understand it.
I have the equation:
$$r \log\left(\frac{n+k}{k}\right)-\log\left(\frac{rn+k}{k}\right)$$
All the variables are real numbers. It is said that the equation above, when $k=0$ or $k \to 0$, is equivalent to the expression:
$$r\log(n) - \log (rn)$$
I do not understand what the above is true. Is this so, and can anybody give me some insight into why this is the case?
Edit: Okay I got an idea, maybe using the fact that $\log(a/b)=log(a)-log(b)$ I can rewrite my expression as
$$ r \log(n+k)+r\log(k)-\log(rn+k)-\log(k)$$
and then with $k=0$
$$ r \log(n)-\log(rn)+(r-1)\log(k)$$
but then I guess the last term is going to $-\infty$ if $r>1$?
Firstly there is a problem with the domain, If they are all real numbers that allows for negative logs which is undefined. It appears that there was either a mistake or the resource in which you found this is incorrect. As far as I can tell, as $k \to 0$ the equation goes to $+/-\infty$ (depending on the value of r), the only time I can see this hold true is when $r=1$ and thus the log(k) terms will cancel.