Investigate the following limit. lim sin. Solution X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
1. Solve and complete
![Example 5
Investigate the following limit.
lim sin()
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.
(²-) =
f(1) = sin(t) = [
f(¹) = sin(3r) = |
f(0.1) = sin(10m) = 0
f(3)
f(0.01)
= sin(2r) = 0
sin ([
sin(100m)
]) = 0
Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim_sin() = [
f(1) = sin(nr) =
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.
=
All
y=sin(w/x)
, but this time our guess is wrong. Note that although
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf9c0948-a07f-432e-b99c-8acd122bf28e%2F2ff8bc77-4a0b-4d80-9f69-127d1bce5d7a%2F5mnow3a_processed.png&w=3840&q=75)
Transcribed Image Text:Example 5
Investigate the following limit.
lim sin()
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.
(²-) =
f(1) = sin(t) = [
f(¹) = sin(3r) = |
f(0.1) = sin(10m) = 0
f(3)
f(0.01)
= sin(2r) = 0
sin ([
sin(100m)
]) = 0
Similarly, f(0.001) = f(0.0001) = 0. On the basis of this information we might be tempted to guess that lim_sin() = [
f(1) = sin(nr) =
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.
=
All
y=sin(w/x)
, but this time our guess is wrong. Note that although
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.
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