Squeeze Theorem Proofs

[yojoe]

Hi-

I'm having trouble writing out the proofs for the following proofs. I'm so tempted to just stick in zero for the theta or x in this case and call it quits, but I know that is not correct. I've tried using certain relationships and that has gotten me nothing but three pages of scribble.

Here are two links for two proofs I'm stuck:


http://img233.imageshack.us/img233/876/cosx12hh.png

http://img225.imageshack.us/img225/2636/sinxx11mt.png


Thx for any help!

 

[yojoe]

For the second proof:

http://img225.imageshack.us/img225/2636/sinxx11mt.png


Both sides need to be proved from both sides, as x-->0+ and x-->0-

So far as I've gotten, I'm thinking that x< sinx/x < 1?

 

[arthur ohlsten]

1)
lim x-->0 cos x = lim x-->0 [1-x^2/2! +x^4/4! + ... ]
lim x--) cos x =1 answer


2)
lim sinx /x= 0/0 undefined by L'hospitals rule
x-->0

lim x-->0 sinx / x = cosx / 1 as x-->0 which is 1/1
1 answer

 

[yojoe]

Hey thanks Arthur-

That should help me figure some things out, but we haven't covered L'hospitals rule yet. Just had a quick question on part 2, so as to prove lim x-->0 sinx / x = 1, all that's needed is just to find the limit? Subbing 0 for x?