Friday, September 14, 2012

The Sandwich (or Squeeze) Theorem

The Sandwich Thm is a very creative way to find limits of functions that are down right impossible to solve using algebriac techniques.  Many times, sandwich thm problems involve trig functions...although they don't have to.  This is because we can easily "start" the problem be sandwiching sine or cosine between 1 and -1...as you no doubt recall from trig. :)

As a motivation let us consider the function

\begin{displaymath}f(x) = x^2 \sin\left(\frac{1}{x}\right)\;\cdot\end{displaymath}



When x get closer to 0, the function $\displaystyle g(x)=\sin\left(\frac{1}{x}\right)$ fails to have a limit. So we are not able to use the basic properties discussed in the previous pages. But we know that this function $\displaystyle g(x)=\sin\left(\frac{1}{x}\right)$ is bounded below by -1 and above by 1, i.e.

\begin{displaymath}-1 \leq \sin\left(\frac{1}{x}\right) \leq 1\end{displaymath}



for any real number x. Since $x^2 \geq 0$, we get

\begin{displaymath}- x^2 \leq x^2\;\sin\left(\frac{1}{x}\right) \leq x^2 \;. \end{displaymath}



Hence when x get closer to 0, x2 and -x2 become very small in magnitude. Therefore any number in between will also be very small in magnitude. In other words, we have

\begin{displaymath}\lim_{x \rightarrow 0}\quad x^2\;\sin\left(\frac{1}{x}\right) = 0\;.\end{displaymath} 
 
 
Enjoy!
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