Investigate the following limit. lim sin x - 0 Solution Notice that the function f(x) = sin(-) is undefined at 0. Evaluating the function for some small values of x, we get the following. f(1) = sin(7) = 0 () = sin(27.) = 0 (= sin (37.) = 0 ( 2 ) = sin (C T. = 0 f(0.1) = sin(107) = 0 f(0.01) = sin(100m) = Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim sin(-) = , but this time our guess is wrong. Note that although f() = sin(nn) = for any integer n, it is also true that f(x) = 1 for infinitely many values of x that approach 0. You can see this from the graph of f given in the following figure. y = sin( 7 /x) The compressed lines near the y-axis indicate that the values of f(x) oscillate between 1 and -1 infinitely often as x approaches 0. (See this exercise.) Since the values of f(x) do not approach a fixed number as x approaches 0, lim sin - does not exist. x -+ 0