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