Is the function y = x * cosx infinity from negative infinity to positive infinity? Why?

Is the function y = x * cosx infinity from negative infinity to positive infinity? Why?

When x → ±∞, y = x * cosx has no limit and is not ∞
xk＝+π/2→+∞ （k∈N.k→+∞）,y＝0→0.
xk＝2kπ→+∞ （k∈N.k→+∞）,y＝xk→+∞.
Note: when x → + ∞, y = x * cosx has no limit and is not + ∞
Similarly, when x → - ∞, y = x * cosx has no limit and is not - ∞

If f (x) = xcosx is bounded on (- ∞, ∞), if f (x) is infinite when x →∞, why?
Please list out a more detailed answer process. Kneel thank you!

F (x) = xcosx is unbounded on (- ∞, ∞). When x = 2npi, xcosx = 2npi tends to infinity
Xcosx = O when x = n + pi / 2
When x →∞, f (x) has no limit

Let y = xcosx, then y '=?

y'=cosx-xsinx