The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to $\mathrm{(A)}\ 2900 \qquad$ $\mathrm{(B)}\ 3000 \qquad$ $\mathrm{(C)}\ 3100 \qquad$ $\mathrm{(D)}\ 3200 \qquad$ $\mathrm{(E)}\ 3300$
Solution
$x$-intercepts of $\sin(1/x)$ occur where $1/x=k\pi$, i.e. $x=\frac1{k\pi}$.
Need $\frac1{k\pi}\in(0.0001,0.001)$, i.e. $k\pi\in(1000,10000)$, so $k\in\left(\frac{1000}\pi,\frac{10000}\pi\right)\approx(318.3,3183.1)$.
Integers $k=319,\dots,3183$: count $=3183-319+1=2865$, closest to (A) $2900$.
