For every integer N>0 consider points:
x_N = Npi-pi/2
they divide the interval [pi/2,+oo) in a set of contiguos intervals such that every point x> pi/2 falls in one of the intervals:
I_N= [x_N, x_(N+1)) = [Npi-pi/2, Npi+pi/2)
at the boundaries of every interval we have:
For N even:
sin (x_N) = sin(2Kpi-pi/2) = sin(-pi/2) = -1
sin (x_(N+1)) = sin(2Kpi-pi/2+pi) = sin(pi/2) = 1
For N odd:
sin(x_N) = sin((2K+1)pi-pi/2) = sin(pi/2) = 1
sin (x_(N+1)) = sin((2K+1)pi-pi/2+pi) = sin((3pi)/2) = -1
In either case for the theorem of intermediate values, as f(x) = xsinx is continuous in the intervals I_N, it assumes in the interval all the possible values between -x_N and x_N.
This means that given any number L and any numbers M,Q > 0 we can find a point bar x > M such that:
abs (f(bar x) - L) > Q
which is in contradiction with f(x) converging to any limit finite or infinite.
graph{xsinx [-191.3, 212.9, -100.5, 101.6]}