Nonexistence of a limit due to oscillation

Plugging in $x = 1, 1/2, 1/3, 1/4, 1/5, 1/6, \dots$ gives $\pi/x = \pi, 2\pi, 3\pi, 4\pi, 5\pi, 6\pi, \dots$, so $s(x) = \cos(\pi/x)$ takes the values $-1, 1, -1, 1, -1, 1, \dots$. Even though these $x$-values get arbitrarily close to $0$, the function values never settle down toward a single number — they keep alternating between $-1$ and $1$ forever.

This is the oscillation failure case: as $x \to 0$, $\pi/x$ grows without bound, so $\cos(\pi/x)$ cycles through all values between $-1$ and $1$ infinitely often no matter how close $x$ gets to $0$. Since $s(x)$ does not approach any single fixed value $L$, the limit does not exist.