Answer
(a) this function is even.
(b) $x=\pm\frac{\pi}{4}, \pm\frac{\pi}{2}, \pm\frac{3\pi}{4}, \pm \pi,\pm\frac{5\pi}{4},... $
(c) The graph of the function is shown in the figure.
(d) As $x\to -\infty, f(x)\to 0$ and as $x\to \infty, f(x)\to 0$
(e) $x\to0, f(x)\to 2$
Work Step by Step
(a) $f(-x)=\frac{sin(-4x)}{-2x}=\frac{sin(4x)}{2x}==f(x)$, so this function is even.
(b) Let $sin(4x)=0, x\ne0$, we can find the x-intercepts as
$x=\pm\frac{\pi}{4}, \pm\frac{\pi}{2}, \pm\frac{3\pi}{4}, \pm \pi,\pm\frac{5\pi}{4},... $
(c) The graph of the function is shown in the figure.
(d) As $x\to -\infty, f(x)\to 0$ and as $x\to \infty, f(x)\to 0$
(e) As $x\to 0$, both the numerator and the denominator approach zero,
and the ratio approaches 2 ($x\to0, f(x)\to 2$) as shown in the graph.