Using the definition of the limit at infinity verify that
0 does not equal lim x-->infinity cosx.
As we know, for any sequence xnwhich converges to infinity, cos(xn) should converge to 0 if it's a limit for cosx. But if we consider the following sequences xn1=pi*n1 and xn2=pi/2+2*pi*n2 we get lim cos(xn1)=1 and lim cos(xn2)=0 1!=0. So 0 does not equal lim cosx x-->infinity.