x EXAMPLE 4 Investigate the following limit. 17T lim sin X x0 sin(1T/x) is undefined at 0. Evaluating the function for some small values of x, we Again the function f(x) SOLUTION get sin(17) f(1) sin(2T) 0 sin(3T) sin 0 - f(0.01) sin(100T) = f(0.1) sin(107) 0 f(0.0001) 0. On the basis of this information we might be tempted to guess that Similarly, f(0.001) 17T lim sin x 0 but this time our guess is wrong. Note that although f(1/n) sin(1n) 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 figure. 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, sini) 17T lim sin does not exist. x0 X

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