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Calculus Level 5

Let \(f : \mathbb{C} \to \mathbb{C}\) be holomorphic on \(\mathbb{C}\).
Also, suppose the following holds: \(\big|f(z)\big| \to +\infty \text{ as } |z| \to +\infty.\)

Then which of the following necessarily holds?

Details and Assumptions:

  • \(\text{Image}(f)\) denotes the range of \(f.\)
  • \(A \setminus B = \{x \in A: \, x \notin B\}.\)
  • If more than one statement is necessarily true, choose the strongest.

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