Solving a Small Limit Problem Without a Calculator | 10x Homework Help

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Homework Help Overview

The discussion revolves around evaluating the limit of the expression (sqrt(cosX) - cosX) / (x^2) as x approaches infinity. Participants are exploring the behavior of the function and its components as x increases without the use of a calculator.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the graphical behavior of the function as x approaches infinity and question the limit's definition. They explore the growth of the denominator and the behavior of the numerator, particularly the implications of cosX as x increases.

Discussion Status

The conversation is active, with participants sharing insights about the limit's behavior. There is an ongoing exploration of the relationship between the numerator and denominator, and some participants are questioning their assumptions about convergence.

Contextual Notes

There is a mention of confusion regarding the limit as x approaches zero versus infinity, and participants are clarifying the conditions under which the limit is being evaluated. The discussion includes considerations of the bounded nature of the cosine function.

alexd
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Homework Statement



lim (x-->infinity) [(sqrt(cosX)-cosX)/(x^2)]

Homework Equations



hey guys, can anyone please help me solve this limit (without a calculator..)

10x!
 
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Welcome to PF!

Hi alexd! Welcome to PF! :smile:

(have a square-root: √ and an infinity: ∞ :wink:)
alexd said:
lim (x-->infinity) [(sqrt(cosX)-cosX)/(x^2)]

Hint: draw the graph …

what does it look like? :smile:

(erm :redface: … you don't mean limx -> 0, do you?)
 
Thanx!

I drew the graph, it seems to be going to zero when x->∞

and unfortunetally it is x->∞ and not x->0

but I'm a bit stuck with the limit...

any ideas on how to approach it (analytically)?
 
alexd said:
I drew the graph, it seems to be going to zero when x->∞

ok … so why is it going to 0? …

how big is the bottom getting? and how big the top? :wink:
 
ok.. I think I see where u're going with this...

so it's actually not defined when x->∞

The case for x->0 is much easier - the answer is 1/4...

is that what u meant?
 
alexd said:
The case for x->0 is much easier - the answer is 1/4...

is that what u meant?

Yes :smile:
ok.. I think I see where u're going with this...

so it's actually not defined when x->∞

Yes it is defined …

just answer my previous question: how do the top and bottom (separately) behave?
 
well...

the bottom obviously goes to ∞ when x->∞

and as for the top.. I have no idea...

that's where i got stuck the 1st time..

what do you do with cosx when x->∞?

the graph is attached
 

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Hi alexd! :smile:

(i can't see your graph yet, but i assume it's upsy-downsy :wink:)
alexd said:
well...

the bottom obviously goes to ∞ when x->∞

Yup! :biggrin:
and as for the top.. I have no idea...

that's where i got stuck the 1st time..

what do you do with cosx when x->∞?

I think you're worrying too much about the fact that it obviously doesn't converge :redface:

but the important point is that it doesn't get out of hand either

|cosx| can't be greater than 1 …

so the top/bottom … ? :smile:
 

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