Express \(\frac{2x^{2}-x+5}{x^{2}-1}\) in the form \(2+\frac{A}{x-1}+\frac{B}{x+1}\), where \(A\) and \(B\) are integers to be found.

The curve \(C\) has equation \(y=\frac{2x^{2}-x+5}{x^{2}-1}\). Show that there are two distinct values of \(x\) for which \(\frac{dy}{dx}=0\).

Sketch \(C\), stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points.

The curve \(C\) has equation \(y=\frac{x+2}{x^{2}-9}\). Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}\lt 0\) at all points on \(C\).

State the equations of the asymptotes of \(C\).

Sketch \(C\), showing the coordinates of any points of intersection with the coordinate axes.

A curve \(C\) has equation \(y=\frac{x^{2}}{x-2}\). Find the equations of the asymptotes of \(C\).

Show that there are no points on \(C\) for which \(0\lt y\lt 8\).

Sketch \(C\), giving the coordinates of the turning points.

The curve \(C\) has equation
\(y=\frac{3 x-9}{(x-2)(x+1)}\)
(i) Find the equations of the asymptotes of \(C\).

(ii) Show that there is no point on \(C\) for which \(\frac{1}{3}\lt y\lt 3\).

(iii) Find the coordinates of the turning points of \(C\).

(iv) Sketch \(C\).