Complex limit involving roots of unity

Solution 1:

If you assume that $g$ and $k$ are analytic at and around $\eta$ then both have a power series expansion valid in a neighbourhood of $\eta$. Because $g(\eta)=k(\eta)$ but $g$ and $k$ are not equal, we know the power series are identical up to the $n$th term and differ in the $n+1$th term for some $n \geqslant 0$. Then we have, \begin{align}g(t) &= a_0+a_1(t-\eta)+a_2(t-\eta)^2+\cdots \\ k(t) &= a_0+a_1(t-\eta)+\cdots+a_n(t-\eta)^n+b_{n+1}(t-\eta)^{n+1}+\cdots \end{align} and as $t \to\eta$, \begin{align} \frac{t^r g(t)-k(t)}{(t^r+1)g(t)-2k(t)} &= \frac{\frac{t^r-1}{t-\eta}a_0 + (t-\eta)^n (t^r a_{n+1}-b_{n+1}) + o((t-\eta)^n)} {\frac{t^r-1}{t-\eta}a_0+(t-\eta)^n ((t^r+1) a_{n+1}-2b_{n+1}) + o((t-\eta)^n)}. \end{align} We can now distinguish three cases: (i) if $a_0=0$ the limit is $1/2$ for all $n \geqslant 0$; (ii) if $a_0 \neq 0$ and $n > 0$ the limit is $1$; and (iii), if $a_0 \neq 0$ and $n=0$ the limit is, \begin{align} \frac{r\eta^{r-1}a_0 + a_1-b_1}{r\eta^{r-1}a_0+2(a_1-b_1)}. \end{align}