complex limit $\lim_{z\rightarrow i}\frac{z^{2}-1}{z^{2}+1}$ exists?

Question

I have a question about this problem, any advice would be greatly appreciated.

Calculate the following limit: $$\lim_{z\rightarrow i}\frac{z^{2}-1}{z^{2}+1}$$

my steps:

I tried factoring the denominator and divisor but I can't eliminate the uncertainty so I think that maybe this limit doesn't exist but I'm not sure about it

$$\lim_{z\rightarrow i}\frac{z^{2}-1}{z^{2}+1} = \lim_{z\rightarrow i}\frac{(z+1)(z-1)}{(z+i)(z-i)}$$

Answer 1

Hint

Let $z=x+i$ and work around $x=0$ $$\frac{z^{2}-1}{z^{2}+1}=\frac{-2 +2ix+x^2} {x(2i+x) }$$