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A small limit problem

  • Thread starter alexd
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  • #1
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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!
 

Answers and Replies

  • #2
tiny-tim
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Welcome to PF!

Hi alexd! Welcome to PF! :smile:

(have a square-root: √ and an infinity: ∞ :wink:)
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?)
 
  • #3
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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)?
 
  • #4
tiny-tim
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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:
 
  • #5
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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?
 
  • #6
tiny-tim
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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?
 
  • #7
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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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  • #8
tiny-tim
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Hi alexd! :smile:

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