If 1≤|f(z)|≤|g(z)||z|−1−ϵ1 \leq |f(z)| \leq |g(z)||z|^{-1-\epsilon} for |z|≥Δ|z| \geq \Delta. Prove that the sum of the residues of fg\frac{f}{g} at all its poles is 00.

Here f and g are entire functions, ϵ,Δ∈(0,∞)\epsilon, \Delta \in (0, \infty).
Clearly the inequality condition says that fg\frac{f}{g} does not have pole for |z|≥Δ|z| \geq \Delta. Then how to show the residues is zero?

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