The limit of product $\prod_{x=2}^k(1-x^{-2})$ as $k\to\infty$

Question

For some reason Wolfram is saying that as $k$ tends to infinity, $\prod_{x=2}^k(1-x^{-2})$ tends to zero, but my book is claiming that this product is never less than one half. Which is true, and why? I can't seem to make any ground into this.

Answer 1

Rewrite $1- x^{-2} = (x^2 - 1)x^{-2}= (x - 1)(x+1)x^{-2}$ and cancel.

Answer 2

$$ \begin{align} \prod_{x=2}^k\frac{x-1}x\frac{x+1}x &=\frac12\overbrace{\boxed{\displaystyle\frac32\cdot\frac23}\boxed{\displaystyle\frac43\cdot\frac34}\frac54\cdots\frac{k-2}{k-1}\boxed{\displaystyle\frac{k}{k-1}\cdot\frac{k-1}k}}^{\text{cancels to $1$}}\frac{k+1}k\\ &=\frac{k+1}{2k} \end{align} $$