test the continuouty of the function f(x)=xSin1/x x#0and f(0)=0=0

Question

And also draw the graph the function and also check the diiffrentibilti

Answer 1

We need to look at the behaviour of the function f(x)=xsin(1/x). When x is very large, 1/x is very small and we can expand sin(1/x)=1/x approx. xsin(1/x)=1 approx as x→±∞. Therefore y=1 is an asymptote. When x is small, 1/x becomes larger but, since the upper and lower limits of sine are -1 and 1, xsin(1/x)→0 as x→0. When x=0, f(0)=0 (given) so the limit of xsin(x) is the same as f(0), and continuity is preserved. As x→0, the amplitude of the sine wave decreases rapidly and so does the period (wavelength), hence the graph above.