Answer
$$0$$
Work Step by Step
Since $-1\leq \cos \frac{1}{t-2}\leq 1$, then we have
$$-(t^2-4)\leq(t^2-4)\cos \frac{1}{t}\leq(t^2-4).$$
Moreover, $\lim\limits_{t \to 2}(t^2-4)=\lim\limits_{t \to 2}-(t^2-4)=0$. Then by the Squeeze Theorem, we have
$$\lim\limits_{t \to 2}(t^2-4)\cos \frac{1}{t-2}=0.$$
