How Do You Prove the Limit of sin(t)/sqrt(t) as t Approaches Infinity?

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Homework Help Overview

The discussion revolves around evaluating the limit of the function sin(t)/sqrt(t) as t approaches infinity. Participants explore the behavior of the sine function at infinity and its implications for the limit.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the application of L'Hopital's rule and question its validity due to the oscillatory nature of sin(t). There is mention of the ε-δ definition for proving limits. The maximum and minimum values of sin(t) are also queried, and the squeeze theorem is suggested as a potential method for evaluation.

Discussion Status

Some participants have offered guidance regarding the squeeze theorem and have prompted further exploration of the limit. There is an acknowledgment of the original problem's complexity, and multiple interpretations of the limit's behavior are being considered.

Contextual Notes

The discussion references a larger problem involving improper integrals, which may influence the approach to the limit in question. There is also a suggestion to consider the entire problem for better context.

Bipolarity
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Homework Statement



\lim_{t→∞}\frac{sin (t)}{\sqrt{t}}

Homework Equations


The Attempt at a Solution


This was actually part of a larger problem about improper integrals. The problem has been reduced to this, but I have no idea how to proceed from here. I know that sin(x) behaves very bizarrely at infinity, so I don't know if L'Hopital's rule can even be applied here.

My intuition tells me that the answer is 0, but how can we prove this? Must we refer to the ε-δ definition?

BiP
 
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What are the maximum and minimum possible values that sin(t) can ever achieve?
 
Bipolarity said:

Homework Statement



\lim_{t→∞}\frac{sin (t)}{\sqrt{t}}

Homework Equations



The Attempt at a Solution


This was actually part of a larger problem about improper integrals. The problem has been reduced to this, but I have no idea how to proceed from here. I know that sin(x) behaves very bizarrely at infinity, so I don't know if L'Hopital's rule can even be applied here.

My intuition tells me that the answer is 0, but how can we prove this? Must we refer to the ε-δ definition?

BiP
Use the squeeze theorem.

What's \lim_{t→∞}\ 1/\sqrt{t}\ ?

How about giving us the entire problem?
 
Are you familiar with the squeeze theorem?
 
Ah, good old squeeze theorem why didn't I think of that?

Thanks guys!

BiP
 
The original problem (for Sammy):

\int^{π}_{0}\frac{dt}{\sqrt{t}+sin(t)}

BiP
 

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