Explain me

EXAMPLE 4 Investigate the following limit. lim sin 5x) x - 0 SOLUTION Again the function f(x) = sin(3x/x) is undefined at 0. Evaluating the function for some small values of x, we get f(1) = sin(3n) = = sin(67) = 0 = sin(97t) = = sin 1 = 0 f(0.1) = sin(30m) = 0 f(0.01) = sin(300m) = Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim sin = x - but this time our guess is wrong. Note that although f(1) = sin(3nn) = 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, lim sin 37 does not exist. X - 0