Looks pretty straight forward: since cos(t) always lies between -1 and 1,
[tex]-1/x\le cos(x^2)/x\le 1/x[/itex]. As x goes to infinity, both -1/x and 1/x go to 0.
As for proving that "the limit of its derivative does not exist", what is the derivative of \(cos(x^2)/x\)?