I made a mistake with \(e\) in \(\mathbb{C}\), but I ammended it

Calculus Level 4

True or false? (In \(\mathbb{C}\)) \[\displaystyle \lim_{z \to 0} (e^{z})^{\frac{1}{z}} = e\]

Note.- We are using the exponentiaton complex, and not the multivalued function or multifunction "\(\text{ exp(z)}\)" ,i.e, if \(h : \mathbb{C} - \{0\} \longrightarrow \mathbb{C} \) is the function \( h(z) = (e^{z})^{\frac{1}{z}} \) then \[\displaystyle \lim_{z \to 0} h(z) = e, \text{ is true or false ?}\] Bonus.- what about \(\displaystyle \lim_{z \to 0} (e^{\frac{1}{z}})^{z} = e\)?

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