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0 $begingroup$ I am reading the book Introduction to Cyclotomic Fields. I found the following limit on page number 33. I have no idea how to obtain the following limit. $$lim_{xto 1-n} (e^{(2 pi i x)}-1)Gamma(x)=frac{(2pi i)(-1)^{(n-1)}}{(n-1)!}.$$ Do we need to use L'Hospital's rule for this? Is there any connection with the residue of Gamma function at negative values? The residue at $z=k$ is given by $operatorname{Re}s_{z=k} Gamma(z)=frac{(-1)^{k}}{k!}.$ Please help me to understand this. complex-analysis number-theory limits gamma-function share | cite | improve this question edited Dec 18 '18 at 21:09