Computing $\lim_\limits{n\to \infty} \log\left(-1+\frac1ni\right)$ and $\lim_\limits{n\to \infty} \log\left(-1-\frac1ni\right)$

Question

Computing $\lim_\limits{n\to \infty} \log\left(-1+\frac1ni\right)$ and $\lim_\limits{n\to \infty} \log\left(-1-\frac1ni\right)$

I am fresman and self learning complex analysis.

ATTEMPT:

When $n$ goes to infity inside log goes to $-1$ in either way. But it is undefined.

So I try module $$\left | \log\left(-1+\frac1ni\right)\right|<??$$

So I have no clue to start?

Answer 1

We have that with reference to the principal value

$$-1+\frac1ni =\sqrt{1+\frac1{n^2}}e^{i(\pi-\arctan\frac1n)} \to e^{i\pi}$$

$$-1-\frac1ni =\sqrt{1+\frac1{n^2}}e^{i(-\pi+\arctan\frac1n)} \to e^{-i\pi}$$

and therefore

$$\log\left(-1+\frac1ni\right)\to i\pi $$ $$\log\left(-1-\frac1ni\right)\to -i\pi $$