1. ## Sandwich Theorem

My teacher just touched on this today. I am trying to figure out some problems. I will show you one, and show how I did it. Could you tell me if I am doing this right? I could be totally off. Thanks in advance.

lim as x approaches infinity of (1-cosx)/(x^2)

-1 <= cos x <= 1

2 <= 1-cosx <= 0

(2/x^2) <= (1-cosx)/(x^2) <= 0/(x^2)

0 <= (1-cosx)/(x^2) <= 0

???

2. Originally Posted by Morgan82
My teacher just touched on this today. I am trying to figure out some problems. I will show you one, and show how I did it. Could you tell me if I am doing this right? I could be totally off. Thanks in advance.

lim as x approaches infinity of (1-cosx)/(x^2)

-1 <= cos x <= 1

2 <= 1-cosx <= 0

(2/x^2) <= (1-cosx)/(x^2) <= 0/(x^2)

0 <= (1-cosx)/(x^2) <= 0

???
$\displaystyle -1 \leq \cos x \leq 1$
$\displaystyle 1 \geq -\cos x \geq -1$

add 1 to all three sides:

$\displaystyle 2 \geq 1 - \cos x \geq 0$

Your reasoning looks okay -- just make sure you justify the way your teacher expects you to.
Good luck!!

3. yup, do take note of your careless mistake. how can f(x) be bigger then 2 and smaller then 0? good luck! =)

1 <= cos x <= 1

2 <= 1-cosx <= 0