Computing the slope of $x^3$ via a limit
\[ \lim_{h\rightarrow 0}\frac{\left(x+h\right)^3-x^3}{h} = 3 x^2 \]

A famous limit of an indeterminate form $0/0$
\[
\lim_{x\rightarrow 0}\frac{\sin{\left(x\right)}}{x} = 1\]

A limit as x approaches infinity
\[
\lim_{x\rightarrow \infty}x^{\frac{1}{x}} =
1\]

A limit involving extra variables
\[
\lim_{x\rightarrow 0}\frac{a^x-b^x}{x} =
\ln{\left(a\right)}-\ln{\left(b\right)}\]

The limit of $1/x$ as $x$ approaches 0 from the left
\[
\lim_{x\rightarrow 0}\frac{1}{x}=
-\infty\]

A limit for which L'Hopital's rule would be slow
\[
\lim_{x\rightarrow 0}\frac{\left(\sin{\left(x\right)}\right)^{249} \left(\ln{\left(1-x\right)}\right)^{251}}{x^{100} \left(\arctan{\left(x\right)}\right)^{400}}=
-1\]

The formula for the slope of $x^3$
\[
\frac{d}{d x} x^3 =
3 x^2\]

An illustration of the chain rule
\[
\frac{d}{d x} \sin{\left(x^3\right)}=
3 x^2 \cos{\left(x^3\right)}\]

A partial derivative
\[
\frac{d}{d y} x^2 y^3=
3 x^2 y^2\]

A 5th-order Taylor polynomial expanded about x=0
\[
\mathrm{TAYLOR}{\left(\mathbf{e}^x,x,0,5\right)}=
0.00833333 x^5+0.0416666 x^4+0.166666 x^3+0.5 x^2+x+1\]

A 7th-order Taylor polynomial
\[
\mathrm{TAYLOR}{\left(\ln{\left(\cos{\left(a x\right)}\right)},x,0,7\right)}=
-0.0222222 a^6 x^6-0.0833333 a^4 x^4-0.5 a^2 x^2\]

An antiderivative of $x^2$ with respect to $x$
\[
\int x^2 dx=
0.333333 x^3\]

An antiderivative of cosine(x) with respect to x
\[
\int \cos{\left(x\right)} dx=
\sin{\left(x\right)}\]

An antiderivative obtained by substitution
\[
\int x^2 \cos{\left(a x^3+b\right)} dx=
\frac{0.333333 \sin{\left(b\right)} \cos{\left(a x^3\right)}}{a}+\frac{0.333333 \cos{\left(b\right)} \sin{\left(a x^3\right)}}{a}\]

A definite integral for x going from a to b
\[
\int_{a}^{b}x^2\:dx=
0.333333 b^3-0.333333 a^3\]

An integral having an infinite integration limit
\[
\int_{a^2}^{\infty}\frac{1}{x^2}\:dx=
\frac{1}{a^2}\]

An integral having an endpoint singularity
\[
\int_{0}^{b^2}\frac{1}{\sqrt{x}}\:dx=
2 \left|b\right|\]

A 2-dimensional integral over a quarter disk
\[
\int_{0}^{r}\int_{0}^{\sqrt{r^2-x^2}}x y\:dy\:dx=
0.125 r^4\]

The formula for the sum of an arithmetic series
\[
\sum_{k=0}^{n} k=
0.5 n \left(n+1\right)\]

The formula for the sum of successive cubes
\[
\sum_{k=0}^{n} k^3=
0.25 n^2 \left(n+1\right)^2\]

The formula for the sum of a geometric series
\[
\sum_{k=0}^{n} a^k=
\frac{a^{n+1}}{a-1}-\frac{1}{a-1}\]

The sum of an infinite series
\[
\sum_{k=0}^{\infty} 2^{-k}=
2\]

A sum for which iteration would be slow
\[
\sum_{k=-123456788}^{123456789} \frac{k}{370370367}=
0.333333\]

The product of successive even integers
\[
\prod_{k=1}^{n} 2 k=
2^n n\]