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\textbf{Useful Limit Theorems.}
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Reminder:
In the following assume all functions have domain some open set and range a subset of the reals.  Assume also that each formula is well-defined (e.g. $p$ is in the common domain where required.)  [There are similar theorems for sequential limits.]

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Definition.  Suppose $f: U \rightarrow \mathbb{R}$ is a function and $p$ is a number in the domain of $f$.  Then
$$\lim_{x \rightarrow p} f(x) = L$$
means that if $\epsilon >0$ then there exists a number $\delta$ so that
\begin{eqnarray*}
|f(x) - L| & < & \epsilon
\end{eqnarray*}
for all $x$ so that
\begin{eqnarray*}
0 & < |x-p| < & \delta.
\end{eqnarray*}

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In the following theorems assume all functions have the number $p$ in their domains.

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Theorem L1.
$$\lim_{x \rightarrow p} (f(x) + g(x)) = \lim_{x \rightarrow p} f(x) + \lim_{x \rightarrow p} g(x).$$

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Theorem L2. If $c$ is a constant then
$$\lim_{x \rightarrow p} (cf(x)) = c \lim_{x \rightarrow p} f(x).$$

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Theorem L3.
$$\lim_{x \rightarrow p} (f(x) \cdot g(x)) = \lim_{x \rightarrow p} f(x) \cdot \lim_{x \rightarrow p} g(x).$$

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Theorem L4.  If
$$\lim_{x \rightarrow p} f(x) \ne 0$$
then
$$\lim_{x \rightarrow p} \frac 1 {f(x)} = \frac 1{\lim_{x \rightarrow p} f(x)} .$$



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