Expert: Steve Holleran - 9/26/2008

Question
Hi I am having a lot of trouble understanding how to use the squeeze theorem.
My equation is  lim x-> 0   : All of the following is under the square root.  (x^2)(cos^2)(1/x)
I really have NO idea how to solve this at all. If you could please help me understand this, that would be great.
Thanks in advance :)

Answer
Hi Steph,

Yeah, the Squeeze Theorem is not exactly an everyday useful result!

I think here you want to think this way:

          sqrt( x^2 * cos(1/x) )  = x * cos(1/x)

Now,  remember that regardless of the argument ( the 1/x ), cosines are always between -1 and 1:

                  -1  <=  cos(1/x)  <= 1

multiply through by x:

                  -x  <= x cos(1/x) <= x

Now, as x --> 0, -x -> 0 and x -> 0 , so the quantity between them

 x cos(1/x) -> 0 also by the S.T.

Hope this helps
Steve