I am aware, that by definition of a slant asymptote, we end up with a limit as $x$ approaches infinity of $\sin(x)/4$, which does not exist, but if we graph $\sin(x) \times (x/4)$, we may see that it approaches and touches $x/4$ and $-x/4$. Are those not asymptotes, or am I missing something? Thank you.
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Generally, an asymptote is a ray that you will stay close to as you travel along the curve. For any $ε > 0$, there is some point on your travel after which you will always be at most $ε$ away from the asymptote. – user21820 Feb 24 '20 at 02:28