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If $f(z)$ is an entire function such that $$ \lim_{x \rightarrow -\infty}\frac{f(x)}{|f(x)|}=1$$ where $|f(x)|$ is the modulus of $f$, and $f(x)$ is just evaluating $f$ at real $x$. What can we say about $f(x)$ (or $f(z)$)?

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Not much. It could be the exponential function, or an even-degree polynomial with positive real leading coefficient, or $\cos z+2$, or $1/\Gamma(-z)$, or ... –  Henning Makholm Oct 24 '11 at 16:08

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