# Sandwich Theorem

• Sep 9th 2009, 05:58 PM
Morgan82
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

(Thinking) ???
• Sep 9th 2009, 06:26 PM
apcalculus
Quote:

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

(Thinking) ???

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

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!!
• Sep 9th 2009, 08:33 PM
guess
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