Finding an analytic function

Question

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I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.

Answer 1

let $z={1 \over n}$, then $f(z) = {{1 \over z} \over {1 \over z}+1} = {1 \over 1+z}$.

Does this suggest anything?

Note that if $f,g$ are analytic on some domain $D$ and agree on a collection of points which have a limit point in $D$ then $f=g$ on $D$.

Where is the pole of this function? How does this limit analyticity?

Note that the question is a little ambiguous; the resulting function will certainly be analytic on $(-1,0)$.