Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide...

Step 1
Substitute for x
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}=\underset{x\to \mathrm{\infty }}{lim}\sin \left(\mathrm{\infty }\right)}$
Simplify
${\displaystyle =Undefined}$
Substitute for x
${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f\left(x\right)=\underset{x\to -\mathrm{\infty }}{lim}\sin (-\mathrm{\infty })}$
Simplify
${\displaystyle =Undefined}$
Step 2
Sine is defined on entire domain so we have no vertical asymptotes.
For slant asymptotes, we check the lines.
Using formula ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{f\left(x\right)}{x}=k}$ where k represents slope of our line (slant asymptote).
Table limit
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{f\left(x\right)}{x}=\underset{x\to \mathrm{\infty }}{lim}\frac{\sin \left(x\right)}{x}}$
Simplify
${\displaystyle =0}$
Therefore, there is no slant asymptote.