Existence of the limit of sin(1/x) as x approaches 0

As $x \to 0$, the quantity $1/x$ grows without bound in magnitude, so it sweeps repeatedly through every real number value, causing $\sin(1/x)$ to oscillate between $-1$ and $1$ infinitely many times no matter how small $|x|$ is. Unlike $d(x) = x\sin(1/x)$, there is no shrinking factor multiplying the oscillation here, so the oscillation is never damped.

Because $e(x)$ keeps cycling through all values in $[-1,1]$ arbitrarily close to $x=0$, it never settles near a single value $L$ as $x \to 0$. This is the oscillation-without-settling case, so $\lim_{x\to 0}\sin(1/x)$ does not exist.