Sample problems; Simple Limit (Epsilon - Delta proofs) Latex code included

[DrWahoo]

Remember to use the appropriate packages; these are in similar post if a mod wants to add the link if you choose to use Latex.

Here is the PDF View attachment 7596

\begin{document}
\begin{center}
	{\LARGE Epsilon-Delta Proofs \\[0.25em]  Practice} \\[1em]
	{\large  Just for practice, don't use Google to cheat!}
\end{center}
\bigskip

\begin{problem}[1.]  Use an $\epsilon$-$\delta$ proof to prove 
		$\ds \lim_{x \to 2} \, \frac{1}{2}  x = 1$.
\end{problem}

\begin{problem}[2.] Use an $\epsilon$-$\delta$ proof to prove 
	$\ds \lim_{x \to -2} \, (-3x + 1) = 7$.
\end{problem} 

\begin{problem}[3.] Use an $\epsilon$-$\delta$ proof to prove 
	$\ds \lim_{x \to 3} \, (6x - x^2) = 9$.
\end{problem} 

\begin{problem}[4.]  Consider the function $f:\RR \to \RR$ defined as 
\[	
	f(x) = \begin{cases} 1 & \text{if $x$ is rational} \\ 
				 0 & \text{if $x$ is irrational.}  \end{cases}
\]
At which values of $x$ is $f(x)$ continuous?  You can assume without proof 
the fact that between every two rationals, there is an irrational and between every two 
irrationals, there is a rational. 

\end{problem}

\begin{problem}[5.]  Give an example of a function $f$ such that $f$ is continuous nowhere, but $|f|$ 
	is continuous everywhere.  \emph{Hint:} Think about problem 4.
\end{problem}\end{document}

 

[I like Serena]

Remember to use the appropriate packages; these are in similar post if a mod wants to add the link if you choose to use Latex.

Hi DrWahoo,

Shouldn't we just include the appropriate packages in the latex code?
That's what for instance https://tex.stackexchange.com requires, so that any documents can be rendered without ambiguity.

 

[DrWahoo]

In the preable you have to dedicate packages to run.
So yes, for this problem some simple packages from ams work.

Most are included in the miktex and other repositories, however, some packages I have made are not on CTAN(mirrors) or in any package manage, so I simply include the "class" or "style" I created in the same directory as my file path so when I choose to compile, it picks up that code.

For instance, I had to make my own thesis style page that had strick guidelines from my graduate school. I had to create my own style and class files to get the format to fit the guidelines. This includes macros and graphing vector fields, and stable manifolds.

I can include some examples in a zip file if anyone wants them.