Find all asymptotes of y=(sin x)/x

Find all asymptotes of $y=\dfrac{\sin x}{x}$.

Step 1: At $x=0$

$\lim_{x\to0}\sin x/x=1$: a removable discontinuity, not an asymptote.

Step 2: As $x\to\pm\infty$

Since $|\sin x|\le1$ for all $x$, \[\lim_{x\to\pm\infty}\frac{\sin x}{x} = 0. \tag{1}\] and $y/x=\sin x/x^2\to0$ as well, so there is no oblique asymptote either — only the horizontal one.

Step 3: Verification

\[x=1000\text{: } y\approx0.000827\to0;\qquad x=-5000\text{: } y\approx-0.0001918\to0\ \checkmark\]