# A small limit problem

## Homework Statement

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

## Homework Equations

10x!

tiny-tim
Homework Helper
Welcome to PF!

Hi alexd! Welcome to PF!

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

Hint: draw the graph …

what does it look like?

(erm … 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)?

tiny-tim
Homework Helper
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?

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?

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

is that what u meant?

Yes
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

#### Attachments

• 1.JPG
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tiny-tim
Homework Helper
Hi alexd!

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

the bottom obviously goes to ∞ when x->∞

Yup!
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

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 … ?