Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 9

The domain of a function is $\mathbb{R}\setminus \{0\}$, because the denominator vanishes at zero. Near zero, $e^x-1\sim x$, so $x=0$ is a vertical asymptote, approached with value $-\infty$ from the left and $+\infty$ from the right. The horizontal asymptotes are $y=-1$ as $x\to -\infty$ and $y=0$ as $x\to +\infty$.

Moreover,Both branches define a strictly decreasing monotone function: the left branch lies below $-1$, and the right branch is positive. There are no zeros or vertical-axis intercept.

The absolute value in the original PDF reflects every negative sine arch upward. The graph is nonnegative, with zeros at $x=k\pi$ for $k=0,1,2,3,4$, and it lies below the decaying envelope $e^{-x}$. On the $k$th arch, differentiation gives a factor proportional to $\cos x-\sin x$. Thus its unique local maximum isSuccessive maximum heights have ratio $e^{-\pi }$. At each interior zero the one-sided slopes have opposite signs, producing a corner. The smaller panels below magnify the later arches so that all four remain visible.