Limit of cos(x)^(1/x) - Solution Ideas

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Homework Help Overview

The discussion revolves around finding the limit of the expression cos(x)^(1/x) as x approaches 0. Participants are exploring the behavior of this limit and the implications of their findings.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants share their initial intuitions about the limit potentially approaching infinity and express confusion regarding the behavior of the function. Some attempt to use Taylor expansion but find it unhelpful. Questions are raised about the tendencies of the numerator and denominator, as well as the implications of signs in the denominator.

Discussion Status

The discussion is ongoing, with participants questioning their intuitions and exploring different interpretations of the limit. There is a recognition of the complexity of the problem, and some participants are seeking clarification on the formulation of the limit itself.

Contextual Notes

There is a mention of potential confusion regarding the correct limit expression, indicating that participants are considering different forms of the limit and their implications.

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Homework Statement



Find the limit:

[tex]\stackrel{lim}{x\rightarrow0}[/tex] cos(x)[tex]^{1/x}[/tex]


The Attempt at a Solution



As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?
 
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a4swe said:

Homework Statement



Find the limit:

[tex]\stackrel{lim}{x\rightarrow0}[/tex] cos(x)[tex]^{1/x}[/tex]


The Attempt at a Solution



As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?

What does the numerator tend to? What does the denomiator tend to?

Also, pay attention to the sign on the denominator. What can happen?
 
Last edited:
a4swe said:

Homework Statement



Find the limit:

[tex]\stackrel{lim}{x\rightarrow0}[/tex] cos(x)[tex]^{1/x}[/tex]


The Attempt at a Solution



As my intuition says this go to infinity this one behaves kind of strange.
My idea for a solution was to use this intuition. Fail.
Second I went to do Taylor-expansion on the cos, which also came up to nothing.

How would you do it?

Your intuition seems to be right, why is it wrong?
If its a stupid question i apologize, I'm working on learning Calculus.
 
Is it supposed to be
[tex]\lim_{x\to 0} \cos(x)^{1/x}[/tex]
or
[tex]\lim_{x\to 0} \frac{\cos(x)}{x}[/tex]
 

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