Disproving the existance of a limit at infinity

Hello

I'm given the function x*sin(x), and asked whether it has a limit of some sort as x goes to negative infinity. In other words does it converge to a number or go off to positive infinity or negative infinity.

Based on the graph of the function my intuition is that it doesn't have a limit of any sort since it is constantly fluctuating between very large and very small values.

I think i've managed to prove that it doesn't converge to any number at negative infinity, because on any interval can always supply an x1 = -pi/2 -n*pi for which the value is -x1 and an x2 = -(3/2) * pi -n*pi for which the value is x2, i.e. the function doesn't get closer to any single number.

But I'm having a lot of trouble with the infinite limits, proving that the function doesn't go off to positive or negative infinity...help would be much appreciated :)