Suppose $N_0 + N_1 = N$ where $N$ is a constant, but $N_i$ can vary and is nonnegative. In my research, I am looking at the value $$\frac{rN_0}{(r-1)N_0+N}, \quad r > 1$$ but not much greater than 1. My question: what is$$\lim_{N \to \infty}\frac{\frac{rN_0}{N}}{\frac{(r-1)N_0+N}{N}}\quad ?$$ If we define $x=N_0/N$, can I say it is $rx$ for small $r$?
You said $N$ is a constant and then asked for a limit as $N\to\infty$. This doesn't make any sense. Fully inform your readers of everything you know about the background of a question; don't expect people to be mind readers. EDIT: At any rate, if $N_0$ is a function of $N$, and $\lim_{N\to\infty} N_0/N\to x$, then the limit you're looking for is $rx/((r-1)x+1)$. –  anon Jul 26 '11 at 2:31
So the $N_i$ vary but $N_0/N$ is kept fixed, at $x$, or at least approaches $x$ as $N$ gets large? We can say the limit is $rx/[(r-1)x+1]$, which indeed is close to $rx$ when $r$ is close to $1$. So close to $rx$ is fine, "it is $rx$" is not right. –  André Nicolas Jul 26 '11 at 2:38
If $r=1+\epsilon$ is close to $1$ then the limit is $x+x(1-x)\epsilon+O(\epsilon^2)$. –  anon Jul 26 '11 at 2:59