What is the Squeeze Theorem and How to Use It in Sequence Calculus Problems?

[rcmango]

Homework Statement

using the squeeze theorem:

lim cosn / sqrt(n)
n -> infinity

Homework Equations

cos/n/sqrt(n) and 1/sqrt(n)

The Attempt at a Solution

I just have a question about the squeeze theorem.

From my understanding, when using the squeeze theorem for these time of sequence calculus problems, I am always going to have the original equation in the middle?

also, why does one side of the squeeze theorem need to be negative?
this sequence approaches 0.

heres the work:
-1/sqrt(n) <= cosn/sqrt(n) <= 1/sqrt(n)

and, if this happened to be sin instead of cosine, would i just put 0 on both sides of the <= instead of the equations.

thanks alot.

 

[mjsd]

it doesn't have to be always negative on the LHS. The idea of the squeeze theorem is to find TWO "simple" functions that you can "squeeze" you function into. Simple in the sense that their limits are easy to evaluate. Of course, you need them to approach the same limit. In this case, you have a negative function -1/sqrt(n) because you know that Cos(n) is bounded by -1 (from below) and +1 (from above).

by the way, in your example n->0, function doesn't approach a finite value.

You need your function in the middle only when you want to find the limit of that function as it approaches some number. In other cases, such as when you just want to find some lower/upper bound (note: may not be the greatest lower or least upper bound) of your function then, you just need to restrict it on one side.

anyway squeezing theorem or sandwich theorem gets its name because you do put your function in the middle when finding limits of complicated function.

 

[rcmango]

thanks for all your help.