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\centerline{MATH 135 -- Calculus 1}
\centerline{The Limit Laws}
\centerline{September 21, 2016}
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\noindent
{\it Background}
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The Limit Laws from section 2.3 and today's video give a number of results that say
we can evaluate limits by breaking down the function involved into simpler pieces.
Provided each piece has a limit separately, we can combine those limits to get 
the limit of a more complicated function. 
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\noindent
{\it Questions}
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\begin{enumerate}
\item[(1)]  Without looking at our book (or any notes you may have taken from today's
video), write out what the Limit Sum, Product, and Quotient Laws tell us in your own words.  
(Then check your work by looking at the big boxed display of Theorem 1 on page 72 of 
our text.)  {\it Comment:}  Especially if you go on to more advanced mathematics
courses in college, one of the things you will be asked to do more and more is
to learn the statements of ``big results'' like this one.  (Eventually you will
need to learn the proofs too!)
\item[(2)]  After completing question (1) and reviewing what those Limit Laws
say, for each of the following limits {\it either} use the Laws to evaluate 
the limit (justifying every step by quoting the Law that applies), or 
else say why the Laws do not apply to that case:
\begin{enumerate}
\item[(a)]  $\lim_{x\to 2} x^3 - 4x^2 + 2x + 7$
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\item[(b)]  $\lim_{x \to 0} \displaystyle{\frac{3^x - 1}{x}}$
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\item[(c)]  $\lim_{x \to 5} \displaystyle{\frac{x^2 + 1}{x^2 + 2x + 4}}$
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\item[(d)]  $\lim_{t \to 4} \displaystyle{\frac{\sqrt{t^2 + 9}}{t^{3/2} + 6}}$
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\item[(e)]  $\lim_{z \to 3} (z^3 - 20)^2$
\end{enumerate}
\item[(3)]  Is it possible for $\lim_{x \to 0} (f(x) + g(x))$ to exist while
$\lim_{x\to 0} f(x)$ and $\lim_{x \to 0} g(x)$ do not exist separately?  If so, give
an example where that happens.  If not, say why not.
\item[(4)]  It definitely is possible for $\lim_{x \to 0} (f(x)\cdot g(x))$ to exist while
$\lim_{x\to 0} f(x)$ and $\lim_{x \to 0} g(x)$ do not exist separately.  Find
an example where that happens.   (Hint: The easiest examples come by looking at piece-wise
defined functions.)
\end{enumerate}
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