Contents
<aside> <img src="/icons/book_orange.svg" alt="/icons/book_orange.svg" width="40px" /> A compendium for Mathematics NS1 written by Martin Deraas has been published on the Amazon store. It is available for purchase as a paperback for £9.95 (120 NOK), and as a PDF document on Google Drive/Adobe Acrobat for NOK 100.
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$$ \lim\limits_{x\to a}f(x)=L $$
Explanation: As the x-axis approaches (but doesn't quite reach) the number a, f(x) gets increasingly close to the limit L.
Sum and difference rule:
$$ \begin{alignedat}{2} \lim\limits_{x\to a}[f(x)+g(x)]&=\lim\limits_{x\to a}f(x)\space &+\space \lim\limits_{x\to a}g(x)\\[5pt] \lim\limits_{x\to a}[f(x)-g(x)]&=\lim\limits_{x\to a}f(x)\space &-\space \lim\limits_{x\to a}g(x) \end{alignedat} $$
Product and constant multiple rule:
$$ \begin{alignedat}{3} \lim\limits_{x\to a}[f(x)\cdot g(x)]&=&\space\lim\limits_{x\to a}f(x)&\cdot \lim\limits_{x\to a}g(x)\\[5pt] \lim\limits_{x\to a}[k\cdot f(x)]&=&k&\cdot \lim\limits_{x\to a}f(x) \end{alignedat} $$
Quotient rule:
$$ \lim\limits_{x\to a}{f(x)\over g(x)}\space={\lim_{x\to a}f(x)\over {\lim_{x\to a}g(x)}}\space {\rm if}\space \lim\limits_{x\to a}g(x)≠0 $$
Power rules:
$$ \begin{alignedat}{5} \lim\limits_{x\to a}[f(x)]&^p&=&&\lbrack\lim\limits_{x\to a}f(x)\rbrack&^p\\ \lim\limits_{x\to a}\lbrack \log_n\space f(x)\rbrack&&=&\space \log_n&\lbrack\lim\limits_{x\to a}f(x)\rbrack \end{alignedat} $$
Exponential rule:
$$ \lim\limits_{x\to a}n^{f(x)}=n^{\lim_{x\to a}f(x)} $$
A conditional function is continuous in the point x=a if:
$$ \lim\limits_{x\to a^-}f(x)=\lim\limits_{x\to a^+}f(x)=f(a) $$
The line x=a is a vertical asymptote of an arbitrary function f if:
$$ \vert f(x)\vert\to\pm\infty\space {\rm when}\space x=a,\space x\to a^+\space{\rm or}\space x\to a^- $$
Alternatively;
$$ \lim\limits_{x\to a^-}f(x)\cup\lim\limits_{x\to a^+}f(x)\cup f(a)=\pm\infty $$
The line x=a is a vertical asymptote of a rational function P(x)/Q(x) if:
$$ Q(a)=0\space ∧\space {P(a)}≠0 $$
The line y=a is a horizontal asymptote of a function f if:
$$ \lim\limits_{x\to ±\infty}f(x)=a $$
Flow chart for calculating limits. Photo: Khan Academy.