A generalization?

True or False?

Let $n_x$ denote the tetration function $n_x = \underbrace{x^{x^{x^{\cdot^{\cdot^{\cdot^ x}}}}}}_{n\text{ number of }x\text{'s}}$ .

Then $\displaystyle \lim_{x\to0^+} n_x = \begin{cases} 0, \quad n \text{ odd} \\ 1, \quad n \text{ even} \end{cases}$ for all integers $n$ .

Bonus: Evaluate $\displaystyle \lim_{x\to\infty} n_x$ .