Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / Section A / 10

The function is nonnegative and satisfies ${\cos }^2(\pi -x)={\cos }^{2}x$. Its derivative is $-\sin (2x)$, so it decreases from $(0,1)$ to its minimum $(\pi /2,0)$ and then increases to $(\pi ,1)$. These three points all have horizontal tangent lines.

The denominator vanishes only at $x=0$, giving a vertical asymptote. Since $1-e^x=-x+O(x^2)$, the function tends to $+\infty$ from the left and $-\infty$ from the right. Its derivative isso each branch is increasing. The horizontal asymptotes are $y=1$ as $x\to -\infty$ and $y=0$ as $x\to +\infty$. The left branch lies above $1$; the right branch lies below $0$, as in the preceding figure.