Now i dont know if i'm right, but maybe my answer can at least point you in the right direction.

Lim(x--> inf+) [sin(2/x)]/(1/x)

= Lim(x--> inf+) (2/2)[sin(2/x)]/(1/x)

= Lim(x--> inf+) 2[sin(2/x)]/(2/x)

Now as x--> infinity, 2/x -->0, so the above is equivalent to:

Lim(a-->0) 2sin(a)/a

= Lim(a-->0+) 2(sin(a)/a)

= 2*Lim(a-->0+) sin(a)/a

= 2*1 ..............................since lim(x-->0) sinx/x = 1

= 2

What bothers me is the whole approaching from the right thing, i dont know how that'll affect my solution