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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Examples of basic limits}

\begin{example}
Let \( f(x)=3x^{2}+1 \). Then \[
\lim _{x\rightarrow 2}f(x)=13.\]

\end{example}

\subsubsection*{Solution }

You can see this most easily by simply plugging into the expression for \( f(x) \)
at \( x=2 \). That ought to be the value of \( f(2) \) since \( f(x) \) has
such a simple expression, and that is the case. The general situation of why
and when you can evaluate some limits just by plugging in are explained in {[}Link:
continuity{]}, and {[}Link: limits-computing{]}

\begin{example}
Let \[
f(x)=\left\{ \begin{array}{ll}
3x+1 & \mathrm{if}\, x\geq 1\\
x+2 & \mathrm{if}\, x<1
\end{array}\right. .\]


Then \[
\lim _{x\rightarrow 1}f(x)\]
 does not exist. 
\end{example}
The reason it doesn't exist is that, as you approach from the left (\( x<1 \)),
the values get towards 3, while from the right (\( x>1) \), the values get
towards 4.  This does have what are called one-sided limits (on either side).
{[}Link to one-sided limits{]}

\begin{example}
Let \[
f(x)=\left\{ \begin{array}{cc}
x^{2} & \mathrm{if}\, x\neq 1\\
5 & \mathrm{if}\, x=1
\end{array}\right. .\]
Then \[
\lim _{x\rightarrow 1}f(x)=1,\]
 even though \( f(1)=5 \). 
\end{example}
Here the function is purposely defined to have the \textit{wrong} value at 1.
The limit \[
\lim _{x\rightarrow 1}f(x)\]
 is what \( f(1) \)  \textit{ought to} be.  

This example stresses the predictability of values of the function. This function
is bad in that it doesn't have the expected value at 1. The previous example
had no value that \( f(1) \) should be; from one side it should be one thing,
from the other side it should be another. But \( f \) has to have just one
value. 

\begin{example}
Let \( f(x)=sin(1/x) \). Then \[
\lim _{x\rightarrow 0}f(x)\]
 doesn't exist.  
\end{example}
This one looks much worse than Example \textbf{2.}. Here, as \( x\rightarrow 0 \),
\( f(x) \) is wildly switching between \( -1 \) and 1.  

\begin{example}
Let \[
f(x)=\frac{x^{3}+1}{x^{2}-1}.\]
 Then \[
\lim _{x\rightarrow -1}f(x)=\frac{-3}{2}.\]
\end{example}

\end{document}
