Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 9 / Solution

Rewrite the rational function as $y=1-1/(x^3+1)$. It is undefined at $x=-1$, with the vertical asymptote $x=-1$, and tends to the horizontal asymptote $y=1$ as $x\to \pm \infty$. At the pole, $y\to +\infty$ from the left and $y\to -\infty$ from the right.

Its derivative isvanishing only at $x=0$. Since $y(0)=0$ and $y=x^3+O(x^6)$ near zero, this is a stationary inflection point, not an extremum. ThusThe graph increases on each side of the vertical asymptote. The left branch lies above $y=1$, while the right branch lies below it and crosses the origin horizontally.