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% Packages
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\usepackage{seqsplit}           % Splits long words.
%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{beabo}
\pdfinfo{
  /Title (math.pdf)
  /Creator (Cheatography)
  /Author (beabo)
  /Subject (math Cheat Sheet)
}

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% This two commands together give roughly
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% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
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% Header and Footer
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\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{math Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{beabo} via \textcolor{DarkBackground}{\uline{cheatography.com/178745/cs/37298/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}beabo \\
  \uline{cheatography.com/beabo} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 25th February, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{LIMITS AND DERIVATIVES}}  \tn
% Row 0
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% Row Count 1 (+ 1)
% Row 1
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% Row Count 2 (+ 1)
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{INTRO}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{We say lim(x→a\textasciicircum{}–\textasciicircum{}) f(x) is the expected value of f at x = a given the values of f near \newline % Row Count 2 (+ 2)
x to the left of a. This value is called the left hand limit of f at a. \newline % Row Count 4 (+ 2)
We say lim (x→a\textasciicircum{}+\textasciicircum{}) f(x) is the expected value of f at x = a given the values of \newline % Row Count 6 (+ 2)
f near x to the right of a. This value is called the right hand limit of f(x) at a. \newline % Row Count 8 (+ 2)
If the right and left hand limits coincide, we call that common value as the limit \newline % Row Count 10 (+ 2)
of f(x) at x = a and denote it by lim(x→a) f(x).% Row Count 11 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{LHL AND RHL}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/beabo_1677298709_Screenshot 2023-02-25 081326.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{1. A constant function takes the same value for all values of x, hence, limit will also be same \newline 2. If value of lhl != rhl, limit is not defined \newline 3. However, at a given point the value of a function and its limit may differ, even when both are defined}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Algebra of limits}}  \tn
% Row 0
\SetRowColor{LightBackground}
Limit of sum of two functions is sum of the limits of the functions & lim x→a {[}f(x) + g (x){]} = lim x→a f(x) + lim x→a g(x) \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
Limit of difference of two functions is difference of the limits of the functions & lim x→a {[}f(x) – g(x){]} = lim x→a f(x) – lim x→a  g(x) \tn 
% Row Count 9 (+ 5)
% Row 2
\SetRowColor{LightBackground}
Limit of product of two functions is product of the limits of the functions & lim x→a {[}f(x) . g(x){]} = lim x→a f(x). lim x→a g(x) \tn 
% Row Count 13 (+ 4)
% Row 3
\SetRowColor{white}
Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero) & lim x→a {[}f(x)/g(x){]} = lim x→a f(x)/lim x→a g(x) \tn 
% Row Count 19 (+ 6)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{In particular as a special case of (iii), when g is a constant function \newline such that g(x) = λ , for some real number λ , we have \newline lim(x→a) {[}( λ.f)(x){]}=λ. lim (x→a) f(x)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Limits of polynomial  functions}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{A function f is said to be a polynomial function if f(x) is zero function or if f(x) = a0 + a1x + a2x2 +. . . + anxn, where aix are real numbers such that an ≠ 0 for some natural number n. \newline % Row Count 4 (+ 4)
1. Iim(x →a)x\textasciicircum{}n\textasciicircum{}=a\textasciicircum{}n\textasciicircum{} \newline % Row Count 5 (+ 1)
2. let f(x)=a0+a1x+a2x\textasciicircum{}2\textasciicircum{}...anx\textasciicircum{}n\textasciicircum{} be a polynomial function. \newline % Row Count 7 (+ 2)
then, f(x)=f(a)% Row Count 8 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}