Answer
$$\lim _{t \rightarrow 0+} \sqrt{t} \cos \left(\frac{1}{t}\right)=0$$
Work Step by Step
Since for $t>0$
$$-1 \leq \cos \left(\frac{1}{t}\right) \leq 1$$
Then
$$-\sqrt{t} \leq \sqrt{t} \cos \left(\frac{1}{t}\right) \leq \sqrt{t}$$
and
$$\lim _{t \rightarrow 0+}-\sqrt{t}=\lim _{t \rightarrow 0+} \sqrt{t}=0$$
Hence, by the Squeeze Theorem, we get
$$\lim _{t \rightarrow 0+} \sqrt{t} \cos \left(\frac{1}{t}\right)=0$$
