Rational function with a finite limit or an oblique asymptote, stationary-point condition and the graph of f of modulus x

Let \(\mathrm{f}(x)=\dfrac{x^2+ax+b}{x-2}\) for \(x\neq2\), where \(a\) and \(b\) are integers.

Suppose first that \(\displaystyle\lim_{x\to2}\mathrm{f}(x)\) exists and is finite.

1. 1. Show that \(b=-2a-4\).
   2. Hence find the value of this limit in terms of \(a\).[4]
2. Suppose instead that the curve \(y=\mathrm{f}(x)\) has a vertical asymptote, has the line \(y=x+5\) as an oblique asymptote, and has at least one stationary point.
   Write down an equation for this vertical asymptote.[1]
3. Show that \(a=3\).[3]
4. Show that \(b\geq-9\).[6]
5. For the case \(a=3\) and \(b=-1\),
   1. sketch the curve \(y=\mathrm{f}(|x|)\), labelling every asymptote with its equation and giving the \(y\)-intercept and the coordinates of each local maximum and local minimum;
   2. solve the inequality \(\mathrm{f}(|x|)<\mathrm{f}(x)\).[7]