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Prazo de entrega: 31 de maio 2022

1. Para quais valores de $k \in \mathbb{R}$ podemos concluir que $$ \forall x \in \mathbb{R} : |\frac{x^2 - kx -1}{x^2 + x +1} | < 3. $$

2. Mostre que $\frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times 2n} < \frac{1}{\sqrt{n}}.$ Claro que isto mostra que

$$ \lim_{n \rightarrow \infty} \frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times 2n} = 0. $$ Qual será $$ \lim_{n \rightarrow \infty} n^{\frac{3}{4}} \frac{1 \times 3 \times \cdots \times (2n-1)}{2 \times 4 \times \cdots \times 2n}? $$