Oscillatory limit failure at x = 1

As $x \to 1$, the denominator $x - 1 \to 0$, so $\dfrac{1}{x-1}$ takes on arbitrarily large positive and negative values very close to $x=1$. Since $\cos(t)$ oscillates between $-1$ and $1$ forever as $t$ ranges over larger and larger values, $\cos\left(\dfrac{1}{x-1}\right)$ cycles repeatedly through all values in $[-1,1]$ infinitely often no matter how close $x$ gets to 1.

This means the function never settles down and approaches a single number as $x \to 1$; instead it keeps oscillating without bound-in-input causing bound-in-output settling. This is the oscillation-type failure of a limit (like $\sin(1/x)$ as $x\to 0$), so the limit does not exist.