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

As $x \to 0$, the quantity $1/x$ increases (or decreases) without bound, so it sweeps through infinitely many multiples of $2\pi$ in any interval around $0$, no matter how small. This means $\sin(1/x)$ repeatedly cycles through every value between $-1$ and $1$ infinitely often as $x$ approaches $0$, rather than settling toward one particular number.

Because $h(x)$ never approaches a single fixed value $L$ — it keeps oscillating between $-1$ and $1$ arbitrarily close to $x=0$ — the limit does not exist. This is the oscillation case: the function fails to settle down near $a=0$, unlike the cases of unequal one-sided limits or unbounded growth.