Question

Using the definition of the limit at infinity, verify that

0 does not equal the limit x-->infinity cosx.

Answer 1

By definition, y0 is a limit off x-->x0 when for any xn-->x0 f(xn)-->y0. now we pick two sequences
xn1=2pi*n1 xn1-->infinity cos(xn1)=1 for any n1. xn2=pi/2+2pi*n2,
xn2-->infinity, but cos(xn2)=0. contradiction.