Solving the Limit of cosx-1/x: Tips & Tricks

[The legend]

Homework Statement

Find the value of
limit x-->0 cosx-1/x

The Attempt at a Solution

one thing i understood is that i can't use the triangle and circle way as i did for
limit x-->0 sinx/x

nor have i found out a way to use the 'sandwich' theorem.
So anyone have any tips please?

I have a guess for it to be 0, but yet, i don't know.

 

[n.karthick]

Is it (cosx-1)/x? Then your guess is correct.

 

[The legend]

yea, but how do i solve it then?

 

[aim1732]

Ever heard of L'Hospital's Rule?

 

[The legend]

nope

 

[aim1732]

Cosine series?I mean expansion of cosx as x-->0?

 

[The legend]

umm...
if x-->0
then,
cosx has limit 1, right?(sorry, but i don't know much calculus.almost none..if you could explain a bit,please )

 

[aim1732]

Alright i will avoid unnecessary calculus. How much trigonometry do you know? Do you know
cos2x = cos²x-sin²x = 2cos²x-1 = 1-2sin²x
Using these formulae a little manipulation and your stated identity sinx/x;x-->0 = 1 try to work out a solution. i will be around if you need help.
Hint:Write cosx as cos2(x/2).

 

[The legend]

yeah i know that trig
so...
using that hint it becomes

=-2sin^2(x/2)/x

= - sin^2(x/2)/(x/2)

= - sin(x/2) * sin(x/2)/(x/2)

but since limit of sin(x/2) as x tends to 0 is 0...hence the whole is 0.

Is this right?

 

[aim1732]

Right on the mark.

 

[The legend]

whew!
Thanks a lot, clever Orange!