"What is the limit of |sin x| as..." was answered by our experts

High School
Calculus and Analysis
Calculus 1
Limits and continuity

Francis Dunlap

Answered question

2023-02-14

What is the limit of |sin x| as x approaches infinity?

Answer & Explanation

ArrillScoorse99z

Beginner2023-02-15Added 4 answers

Consider what happens as x increases without bound.
For any value of x, no matter how great, there are values of the form

π n

and

\frac{π}{2} + π n

to the right of x.
Thus, there are places to the right where

| \sin x | = 0

and places to the right where

| \sin x | = 1

.
The values of |sin x| cannot be getting closer and closer to a single value.
A more rigorous argument can be based on the following.
If someone claims that the limit is L, they must show that for every positive

ε

, there is a number M such that for all

x > M

, we have

| | \sin x | - L | < ε

I claim that for

ε \leq \frac{1}{2}

this is not possible.

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