Consider the function f(x)=sin(1/x)

Question

Find a sequence of x-values that approach 0 such that

sin (1/x)=0
sin (1/x)=1
sin (1/x)=-1

Is sin sin (1/x)=0 and sin (1/x)=-1 does not exist.

What is sin (1/x)=1 then.

Answer 1

sin (1/x) = 0 if 1/x = pi, which means x = 1/pi

sin (1/x) = 1 if 1/x = pi/2, which means x = 2/pi

sin (1/x ) = -1 if 1/x = 3 pi/2, which means x = 2/(3 pi)

I don't understand why the x values should approach zero, or what the rules of the sequence are.