A function is defined by \(\mathrm f(x)=\arccos\left(\dfrac{x^2-1}{x^2+1}\right)\), \(x\in\mathbb R\).
Show that \(\mathrm f\) is an even function.[1]
By considering limits, show that the graph of \(y=\mathrm f(x)\) has a horizontal asymptote and state its equation.[3]
For \(x\in\mathbb R\), \(x\neq0\), show that \(\mathrm f'(x)=-\dfrac{2x}{|x|(x^2+1)}\).[5]
Hence, show that \(\mathrm f(x)\) is increasing for \(x<0\).[2]
Using L’Hôpital’s rule, find \(\displaystyle\lim_{x\to\infty}(x\mathrm f(x))\).[5]