EXAMPLE 4 Investigate the following limit. lim sin SOLUTION Again the function f(x) = sin(57/x) is undefined at 0. Evaluating the function for some small values of x, we get f(1) = sin 5n = 0 () = sin 10n = 0 = sin 157 =0 = sin 20 TT = 0 f(0.1) = sin 50n = 0 f(0.01) = sin 500n = 0 Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim sin 5n = 0 but this time our guess is wrong. Note that although f(1/n) = sin 5nm = X 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. (Use a graphing device to graph f(x) and zoom in toward the origin several times. What do you observe?) Since the values of f(x) do not approach a fixed number as x approaches 0, lim sin does not exist.

EXAMPLE 4 Investigate the following limit. lim sin SOLUTION Again the function f(x) = sin(57/x) is undefined at 0. Evaluating the function for some small values of x, we get f(1) = sin 5n = 0 () = sin 10n = 0 = sin 157 =0 = sin 20 TT = 0 f(0.1) = sin 50n = 0 f(0.01) = sin 500n = 0 Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim sin 5n = 0 but this time our guess is wrong. Note that although f(1/n) = sin 5nm = X 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. (Use a graphing device to graph f(x) and zoom in toward the origin several times. What do you observe?) Since the values of f(x) do not approach a fixed number as x approaches 0, lim sin does not exist.

Name: x marks the spot EXAMPLE 4Investigate the following limit.lim sinSOLUT
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Description: x marks the spot EXAMPLE 4Investigate the following limit.lim sinSOLUTIONAgain the function f(x) = sin(5t/x) is undefined at 0. Evaluating the function for some small values of x, we get()()f(1)= sin 5n =0= sin 10n = 0= sin 15n = 0= sin 20TT = 0f(0.1

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter1: Functions

Section1.2: Functions Given By Tables

Problem 32SBE: Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it...

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