A small limit problem

[alexd]

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!

 

[tiny-tim]

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 lim_{x -> 0}, do you?)

 

[alexd]

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]

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? ​

 

[alexd]

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]

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?

 

[alexd]

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

 

[tiny-tim]

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 … ?