How do you get the limit of x sin (1/x) as x approaches positive infinity?

As my colleague suggested, multiply the numerator and denominator by 1/x
Lim x >>inf xSin(1/x) = Lim x >>inf x(1/x)sin(1/x)/(1/x)
= Lim x>>inf sin(1/x)/(1/x) = Lim a>>0 sin(a)/a
This form of the limit now satisfies l'Hopital's rule's conditions
So your original limit is equal to Lim a>>0 cos(a)/1 = cos(0) = 1

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