The textbook asks me to verify whether or not this statement is true:
The limit of $Log(-1 + \frac{i}{n}) + Log(-1 - \frac{i}{n})$ as $n \rightarrow \infty$ is $0$, given that $Log(z)$ is the main branch of $log(z)$ . So my approach was the following:
$Log(z)$ is not continious for ($y=0$,$x \leq 0$), so if we define $z_1(n) = -1 +\frac{i}{n}$ and $z_2(n) = -1 - \frac{i}{n}$ then it's clear that $z_1(n) \overset{n \rightarrow \infty}{\rightarrow} -1$ and $z_2(n) \overset{n \rightarrow \infty}{\rightarrow} -1$. So given that -1 lies on the branch cut it's correct to say that i can't use the following property: $Log(z_i(n)) \overset{n \rightarrow \infty}{\rightarrow} Log(-1)$ because $Log(z)$ is not continious on that point ? If that's the case how can i evaluate this limit?