Question

evaluate lim of x approaching infinity of 2xsin(1/x)

Answer 1

You can use this limitation \[\lim_{x \rightarrow 0}\frac{ \sin x }{ x } = 1.\]
Let \[y = \frac{ 1 }{ x }\] then, your limitation becomes \[2\lim_{y \rightarrow 0} \frac{ \sin y }{ y } = 2\times1 = 2\]