# Sandwich Theorem

• Oct 26th 2008, 08:27 AM
Caity
Sandwich Theorem
Use the Sandwich/Squeeze Theorem to prove that

$\displaystyle \lim_{n\rightarrow \infty} 2^{-n}\cos(n^3-n^2-13)$

exists.

What is the limit?
• Oct 26th 2008, 08:33 AM
ThePerfectHacker
Quote:

Originally Posted by Caity
Use the Sandwich/Squeeze Theorem to prove that

$\displaystyle \lim_{n\rightarrow infinity} 2^{-n}\cos(n^3-n^2-13)$

exists.

What is the limit?

$\displaystyle |2^{-n} \cos (n^3 -n^2 -13) | \leq 2^{-n} \to 0$
• Oct 26th 2008, 08:34 AM
Moo
Hello,
Quote:

Originally Posted by Caity
Use the Sandwich/Squeeze Theorem to prove that

$\displaystyle \lim_{n\rightarrow infinity} 2^{-n}\cos(n^3-n^2-13)$

exists.

What is the limit?

Use the fact that $\displaystyle -1 \leq \cos(x) \leq 1 \quad \text{for all x}$
• Oct 26th 2008, 08:38 AM
Caity
I did that Moo... then I got stuck... I have a problem saying it all... I was also wondering if I needed to use a subsequence to prove it...
• Oct 26th 2008, 08:40 AM
Moo
Quote:

Originally Posted by Caity
I did that Moo... then I got stuck... I have a problem saying it all... I was also wondering if I needed to use a subsequence to prove it...

Note that $\displaystyle 2^{-n}=\frac{1}{2^n} \to 0$

$\displaystyle -2^{-n} \leq 2^{-n} \cos(\dots)\leq 2^{-n}$

Finish it off :)