Exploring the Limit of sin 1/x as x→0

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In summary, the limit of sin(1/x) as x approaches 0 can be thought of as the same as the limit of sin(x) as x approaches infinity. However, since these limits do not exist, the distinction between left and right approaches is not significant.
  • #1
Echo 6 Sierra
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Either I can't find the correct la text or I'm doing it wrong but here goes:

for a limit where x-->0 for sin 1/x

Am I just supposed to recognize that as x goes to zero from the left that it goes to negative infinity and as x goes to zero from the right it goes to positive infinity? What else could I deduce from this?
 
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  • #2
the problem is same as
[tex] \lim_{x \rightarrow \infty} sin(x) [/tex]
the limit will oscillate from -1 to 1
(not all limit is define)
 
  • #3
Thank you.
 
  • #4
For [tex] \lim_{x \rightarrow 0} sin \frac{1}{x} [/tex] could you please explain how it is the same as[tex] \lim_{x \rightarrow \infty} sin(x) [/tex] ? I understand that not all limits can be defined but is it oscillating between -1 and 1 because it can't be defined?
 
  • #5
You need to specify how that limit goes to zero.Either "goes down or up"...

[tex] \lim_{x\nearrow 0} [/tex] or [tex] \lim_{x\searrow 0} [/tex]

Daniel.
 
  • #6
[tex] \lim_{x \rightarrow 0^+} sin1/x = \lim_{u \rightarrow {+ \infty}} sin u [/tex]
[tex] \lim_{x \rightarrow 0^-} sin1/x = \lim_{u \rightarrow {- \infty}} sin u [/tex]

substitude u=1/x and you will see why
sory for the sloppy notation in my first post
 
  • #7
However, since none of those limits exist, the distinction is moot.
 

What is the limit of sin(1/x) as x approaches 0?

The limit of sin(1/x) as x approaches 0 is undefined. As x gets closer and closer to 0, the value of sin(1/x) oscillates between -1 and 1, never approaching a specific value.

Why is the limit of sin(1/x) as x approaches 0 undefined?

The function sin(1/x) is undefined at x=0 because the denominator becomes 0, resulting in an indeterminate form. Therefore, the limit cannot be determined using traditional methods.

Can the limit of sin(1/x) as x approaches 0 be calculated using L'Hôpital's rule?

Yes, L'Hôpital's rule can be used to calculate the limit of sin(1/x) as x approaches 0. By taking the derivative of both the numerator and denominator, the limit can be rewritten in a form that can be evaluated.

What is the relationship between the limit of sin(1/x) as x approaches 0 and the graph of the function?

The limit of sin(1/x) as x approaches 0 is related to the graph of the function by the vertical asymptote at x=0. As x gets closer and closer to 0, the function approaches the vertical asymptote but never touches it, creating an undefined or infinite value.

Are there any real-world applications for understanding the limit of sin(1/x) as x approaches 0?

Yes, the limit of sin(1/x) as x approaches 0 is used in various fields of science and engineering, such as signal processing, quantum mechanics, and electrical engineering. It is also used in the study of oscillatory and periodic phenomena in nature.

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