%;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

\[
\prod_{k=1}^{n} 2 k\]

%
\[
2^n n\]