IB AA HL May 2026 P2 TZA Q12

The function \(f\) is defined as \(f(x)=\frac{x^2+ax+b}{x+1}\), where \(x\ne-1\) and \(a,b\in\mathbb{Z}\).

First consider the case where \(\displaystyle\lim_{x\to-1}f(x)\) is finite.

1. 1. Show that \(b=a-1\).
   2. Find \(\displaystyle\lim_{x\to-1}f(x)\) in terms of \(a\).[4]
2. Now consider the case where the graph of \(f\) has a vertical asymptote, an oblique asymptote with equation \(y=x-4\), and at least one point of zero gradient.
   Write down the equation of the vertical asymptote.[1]
3. Show that \(a=-3\).[3]
4. Show that \(b\geq-3\).[6]
5. In the case where \(a=-3\) and \(b=1\)
   1. sketch the graph of \(y=f(|x|)\), showing any asymptotes with their equations, the value of the intercept with the \(y\)-axis, and the coordinates of any local maximum or minimum points;
   2. solve \(f(|x|)<f(x)\).[7]