# Answer to Question #18589 in Real Analysis for Matthew Lind

Question #18589

Using the definition of the limit at infinity, verify that

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

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

Expert's answer

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.

xn1=2pi*n1 xn1-->infinity cos(xn1)=1 for any n1. xn2=pi/2+2pi*n2,

xn2-->infinity, but cos(xn2)=0. contradiction.

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