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\lhead{\color{blue} \Large Lecture Notes}
\chead{\color{black} \LARGE Properties of limits}
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\begin{center}
{\LARGE Limit Laws \ \bigskip }
\end{center}

If $L$, $M$,\ and $c$ are real numbers and \ $\lim\limits_{x\rightarrow
\infty }f\left( x\right) =L$ and $\lim\limits_{x\rightarrow \infty }g\left(
x\right) =M$, then\bigskip 

\begin{enumerate}
\item constant rule: $\lim\limits_{x\rightarrow \infty }c=c$

\item identity function rule: \ $\lim\limits_{x\rightarrow \infty }x=\infty $

\item sum rule: $\ \ \ \lim\limits_{x\rightarrow \infty }\left( f\left(
x\right) +g\left( x\right) \right) =L+M$

\item difference rule: $\ \ \ \ \ \ \lim\limits_{x\rightarrow \infty }\left(
f\left( x\right) -g\left( x\right) \right) =L-M$

\item constant multiple rule \ \ $\lim\limits_{x\rightarrow \infty }\left(
c\cdot f\left( x\right) \right) =c\cdot \lim\limits_{x\rightarrow \infty
}f\left( x\right) =c\cdot L$

\item product rule $\ \ \ \ \ \ \ \lim\limits_{x\rightarrow \infty }\left(
f\left( x\right) \cdot g\left( x\right) \right) =L\cdot M$

\item quotient rule \ If $M\not=0$ then $\ \ \ \ \ \
\lim\limits_{x\rightarrow \infty }\dfrac{f\left( x\right) }{g\left( x\right) 
}=\dfrac{L}{M}$

\item power rule $\ \ \ \ \ \ \ \ \ \ \lim\limits_{x\rightarrow \infty
}\left( f\left( x\right) \right) ^{n}=L^{n}$ \ \ \ \ \ \ \ \ \ \ $n\in 
%TCIMACRO{\U{2115} }%
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\mathbb{N}
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$

\item root rule $\ \lim\limits_{x\rightarrow \infty }\sqrt[n]{f\left(
x\right) }=\sqrt[n]{L}=L^{1/n}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ \ 

If $n$ is even, we assume that \ $\lim\limits_{x\rightarrow \infty }f\left(
x\right) =L\geq 0$.

\item replacement rule: \ If the functions $t$ and $s$ have the same values
for all but finitely many values of $x$ , \ then $\lim\limits_{x\rightarrow
\infty }t\left( x\right) =\lim\limits_{x\rightarrow \infty }s\left( x\right) 
$
\end{enumerate}

\bigskip 

\bigskip 

\bigskip 

\bigskip 

Note that all statements are true for limits as $x\rightarrow -\infty $

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