# Think this is easy?

Calculus Level 5

Define $$g_n (t) = e^{t + e^{t + e^{t + \cdots}}}$$, $$n$$ times. And $$\displaystyle f(t) =\lim_{n\to\infty} g_n (t)$$. Then which of the following is/are true?

(a): $$f : \mathbb R^- \to \mathbb R^+$$.
(b): $$f(-1) = 1$$.
(c): $$f$$ is one-to-one.
(d): $$f$$ is increasing on $$\mathbb R$$.
(e): $$f$$ exists for all $$t \in \mathbb R$$.
(f): $$f$$ is increasing on $$\mathbb R^-$$ excluding $$(-1,0)$$.

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