Estimating lim sin(1/x) as x -> 0 from a graph

Looking at the behavior of $g(x) = \sin\left(\dfrac{1}{x}\right)$ near $x=0$: as $x$ approaches $0$ from either side, the argument $1/x$ grows without bound in magnitude, causing $\sin(1/x)$ to cycle repeatedly through all values between $-1$ and $1$. No matter how small a neighborhood around $x=0$ you consider, the function continues to oscillate rapidly through its full range rather than leveling off toward a single height.

Since $g(x)$ does not approach any single fixed value $L$ as $x \to 0$ — it keeps oscillating between $-1$ and $1$ infinitely often — the limit does not exist. This is the oscillation-type failure of a limit, distinct from the function blowing up or the two one-sided limits disagreeing.