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\centerline{MATH 135 -- Calculus 1}
\centerline{The Squeeze Theorem and Trigonometric Limits}
\centerline{October 2, 2019}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
In today's video, we saw an additional technique for evaluating limits called the ``Squeeze Theorem''
and the very important limit:
\begin{equation}
\label{sinlimit}
\lim_{\theta\to 0} \frac{\sin(\theta)}{\theta} = 1.
\end{equation}
Recall that the Squeeze Theorem says:  Assume $l(x) \le f(x) \le u(x)$ on some interval
containing $x = c$ (except possibly at $x = c$) and $\lim_{x\to c} l(x) = \lim_{x\to c} u(x) = L$ (we might
say ``$f$ is squeezed by $l$ and $u$ at $x = c$'' to describe this).  Then
$\lim_{x \to c} f(x)$ exists and equals $L$ as well.   
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\noindent
Do the following problems from Section 2.6 in our text:
\begin{enumerate}
\item[(1)]  Suppose the graphs $y = f(x), y = l(x), y = u(x)$ are as in the plot on the back.
What can we say about $\lim_{x\to 1} f(x)$?
\item[(2)]  Determine $\lim_{x\to 0} f(x)$ given that $\cos(x) \le f(x) \le 1$ for all $x$.
\item[(3)]  State whether the given inequality provides sufficient information to determine
$\lim_{x\to 1} f(x)$, and if so, find the limit.  (Hint:  Draw pictures!)
\begin{enumerate}
\item[(a)] $4 x - 5 \le f(x) \le x^2$
\item[(b)] $2x - 1 \le f(x) \le x^2$
\item[(c)] $4x - x^2 \le f(x) \le x^2 + 2$
\end{enumerate}
\end{enumerate}
\noindent
Evaluate the following limits using \eqref{sinlimit}:
\begin{enumerate}
\item[(4)] $$\lim_{t\to 0} \frac{\sin(t)}{8t}.$$
\item[(5)] $$\lim_{t\to 0} \frac{\sin(8t)}{t}.$$
(Hint:  For this one let $u = 8t$ and convert the $t$ in the denominator to an equivalent expression
in terms of $u$.  Note that $t \to 0$ implies $u = 8t \to 0$ also.)
\item[(6)] $$\lim_{t\to 0} \frac{\sin(3t)}{\sin(5t)}.$$
(Hint:  Rewrite as follows:
$$\frac{\sin(3t)}{\sin(5t)} = \frac{\frac{\sin(3t)}{t}}{\frac{\sin(5t)}{t}},$$
then proceed as in question (5).
\item[(7)]  $$\lim_{t\to 0} \frac{1 - \cos(t)}{t^2}$$
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{Squeeze.png}
\caption{Plot for Question 1, $y = l(x), u(x)$ in blue; $y = f(x)$ in red.}
\end{center}
\end{figure}
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