Lim of sqrt(x) * sine(1/sqrt(x)): Solving for x --> infinity

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Discussion Overview

The discussion revolves around evaluating the limit of the expression sqrt(x) * sine(1/sqrt(x)) as x approaches infinity. Participants explore the behavior of the sine function as its argument approaches zero and the implications for the limit's value.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that as x approaches infinity, 1/sqrt(x) approaches zero, leading to the conclusion that sine(1/sqrt(x)) approaches 1, thus suggesting the limit is infinity.
  • Another participant questions the initial assumption about sine(0), clarifying that sin(0) equals zero, not one.
  • Several participants mention the concept of an indeterminate form in this limit evaluation.
  • Some participants suggest applying L'Hopital's rule, while others argue that a simple rearrangement of terms suffices for solving the limit.
  • There is a discussion about using a substitution (u = 1/sqrt(x)) to analyze the limit further, with one participant expressing curiosity about this method.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to evaluate the limit, with some advocating for L'Hopital's rule and others preferring a rearrangement of terms. There is no consensus on the best method or the final value of the limit.

Contextual Notes

Participants mention the indeterminate form and the need for careful handling of limits, but specific mathematical steps and assumptions remain unresolved.

Starcrafty
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I am doing a question that goes like this Lim of sqrt(x) * sine(1/sqrt(x)) as x --> infinity = ? what i determined was as x --> infinity 1/sqrt(x) would approach zero there for sine of 1/sqrt(x) would approach 1 there fore 1 * sqrt(infinity) would be infinity. however the answer says it is 1. can sum1 explain this?
 
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What is sin(0)?
 
Starcrafty said:
I am doing a question that goes like this Lim of sqrt(x) * sine(1/sqrt(x)) as x --> infinity = ? what i determined was as x --> infinity 1/sqrt(x) would approach zero there for sine of 1/sqrt(x) would approach 1 there fore 1 * sqrt(infinity) would be infinity. however the answer says it is 1. can sum1 explain this?

d_leet said:
What is sin(0)?
That wouldn't explain why the limit is 1 rather than 0!

Starcrafty, you've probably already proved that sin(x)/x goes to 1 as x goes to 0. What happens if you let u= 1/\sqrt{x}?
 
HallsofIvy said:
That wouldn't explain why the limit is 1 rather than 0!

I never claimed it would, but reading the original post the OP seems to have the idea that sin(0)=1.
 
Sorry, I see your point now!
 
Yes, what you actually have is an indeterminate form. As they said above, as x -> infinity, 1/sqrt(x) goes to 0. The sin(0) is not 1, but zero. Therefore you should try applying L'Hoppital's rule (theorem?).
 
Ignea_unda said:
Yes, what you actually have is an indeterminate form. As they said above, as x -> infinity, 1/sqrt(x) goes to 0. The sin(0) is not 1, but zero. Therefore you should try applying L'Hoppital's rule (theorem?).
L'Hopital's is not required here. A simple rearrangement of terms and proper use of limits will do.
 
Last edited:
neutrino said:
L'Hopital's is not required here. A simple rearrangement of terms and proper use of limits will do.

Please show, at least how to set it up, because I am not seeing the rearrangement of the terms.
 
Then go back and read my first response!
 
  • #10
neutrino said:
L'Hopital's is not required here. A simple rearrangement of terms and proper use of limits will do.

I agree that L'Hopital's rule is always easiest for limits that can support it. It is my main tool for all limits (excluding multivariable ones and limits that require other forms of solving and proof).
 
  • #11
HallsofIvy said:
That wouldn't explain why the limit is 1 rather than 0!

Starcrafty, you've probably already proved that sin(x)/x goes to 1 as x goes to 0. What happens if you let u= 1/\sqrt{x}?

I'm sorry I'm being a pain, but what are you meaning "u" to be. Are you replacing 1/x with 1/\sqrt{x}?
 
  • #12
Ignea_unda said:
I'm sorry I'm being a pain, but what are you meaning "u" to be. Are you replacing 1/x with 1/\sqrt{x}?

He is meaning u to be 1/\sqrt{x}, what happens if you make this substitution? What does the limit become in terms of u?
 
  • #13
Hmmm, that's an interesting way to compute the limit. I've never seen it done with u-substitution before.
 
  • #14
Ignea_unda said:
I've never seen it done with u-substitution before.
You would have, if you had checked your PM Inbox. :wink:
 

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