0es ex: show that the function f, defined on r{0} by f(x) = sin(1/x), whenever x # 0, doe approach 0 as x - 0. sol. let & =>0 and let o be any positive number.

0es Ex: Show that the function f, defined on R{0} by f(x) = sin(1/x), whenever x # 0, doe approach 0 as x - 0. Sol. Let & =>0 and let o be any positive number. By Archimedean property, for 1/6 > 0, 3n e N such that 1 - or 2nT+T/2 2nT+ <ö. 1 X then 0< |x 0 <8. 2nT+T/2 Now Isin (1/x) - 0 = |sin (2n7 +/2) = 1> E. Thus there is an e =>0, such that for each ð > 0, 3 some point x = 1 for whi 2nT+T/2' sin (1/x) - 0> E and 0 < |x - 0| < 8. Hence lim sin (1/x) 0. X