Is there an entire function which is not a polynomial such that $\lim_{z\to\infty}\frac{f\left(z\right)}{z}=\infty $ I'm wondering if there's an entire function $f$ which is not a polynomial such that $\lim_{z\to\infty}\frac{f\left(z\right)}{z}=\infty$?
Thanks in advance!

Your hypotheses imply that $f(z)/z$ is everywhere meromorphic on the Riemann sphere, and is thus a rational function.
The only affine singularity is at zero, so the denominator of $f(z) / z$ must be a power of $z$.
But because $f(z)$ is entire, the denominator of $f(z) / z$ cannot have more than one factor of $z$, and thus $f(z)$ is a polynomial.
