I was hoping that Bobosharif S. would have had a shot at this problem. Since he hasn't, I'll try to point you in a possibly correct direction although I have not worked out all the ε δ formalities to be sure.
If we consider x small enough (sin x)/x can be made arbitrarily close to 1 - x2/6 and then
the [sin(x)/x]1/x can be expanded by the binomial theorem to 1 - (x/6) + x3(1 -x)/12 and terms of higher order
Therefore, x can again be made small enough that the given function will be as close as required to 1.
Sorry, not an entirely formal or satisfactory proof, but it could give you a start. If you get a better answer, either here or from your instructor, I would appreciate knowing it.
