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

• LaTeX
• DrWahoo
In summary, the conversation discusses the use of appropriate packages for latex code and the possibility of including custom-made styles and classes for specific formatting needs. The conversation also mentions the inclusion of examples in a zip file for anyone who may be interested.
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
Code:
\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}

#### Attachments

• hw2new.pdf
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DrWahoo said:
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.

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.

## What is a sample problem for a simple limit?

A sample problem for a simple limit is finding the limit of the function f(x) = x as x approaches a specific value, such as 2. This can be represented as lim x→2 x.

## What is an epsilon-delta proof?

An epsilon-delta proof is a method used to rigorously prove the existence of a limit. It involves finding a value, δ, such that if the distance between the input, x, and the limit, L, is less than δ, then the distance between the output, f(x), and the limit, L, is less than a given value, ε. This can be represented as |x-L| < δ → |f(x)-L| < ε.

## How do you write an epsilon-delta proof in LaTeX?

To write an epsilon-delta proof in LaTeX, you can use the following format:
\lim_{x \to c} f(x) = L means for all ε > 0, there exists a δ > 0 such that if |x-c| < δ, then |f(x)-L| < ε. This translates to "the limit of f(x) as x approaches c is L, where for all ε greater than 0, there exists a δ greater than 0 such that when the distance between x and c is less than δ, the distance between f(x) and L is less than ε."

## What is the importance of epsilon-delta proofs?

Epsilon-delta proofs are important because they provide a rigorous and precise way to prove the existence of a limit. They also allow for a better understanding of the behavior of a function near a specific value, and they can be used to prove important theorems in calculus.

## What are some common mistakes when solving sample problems for simple limits?

Some common mistakes when solving sample problems for simple limits include not properly simplifying the function, using the incorrect limit notation, not considering the behavior of the function at the specific value, and not fully understanding the concept of limits. It is important to carefully check the solution and make sure it follows the rules and principles of limit calculations.

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