matma.tex

\documentclass[12pt]{article}
\usepackage[MeX, nomathsymbols]{polski}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{amsmath} %pakiet matematyczny
\usepackage{amssymb} %pakiet dodatkowych symboli
\begin{document}
\texttt{Wyrażenie \#1}
$$\lim_{n\to \infty} \frac{n+1}{n}=1$$
$$\lim_{n\to\infty} \frac{(-1)^n}{n}=0$$
$$\lim_{n\to\infty} \frac{2n+5}{n}=2$$
$$\lim_{n\to\infty} (2n-1)=\infty$$
$$\lim_{n\to\infty} (-n^2+1)=-\infty$$
\texttt{Wyrażenie \#2}
$$
\begin{cases}
|z| = |z-4i| \\
\frac{\pi}{4} \leq \operatorname{Arg}z < \frac{\pi}{2}
\end{cases}
$$
$$
\begin{cases}
|z+4|=|z+2-2i| \\
|z| \leq 2
\end{cases}
$$
$$
\begin{cases}
|z-1-i|<\sqrt{2} \\
\operatorname{Arg}(z-1-i)<\frac{\pi}{2}
\end{cases}
$$
\texttt{Wyrażenie \#3}
$$
\left\{
\begin{array}{rcrcr}
x & + & 5y& = & 2\\
-3x & + & 6y & = & 15
\end{array}
\right.
$$
$$
\left\{
\begin{array}{rcrcrcr}
x& - & y & - & z & = & 1 \\
3x& + & 4y & - & 2z & = & -1 \\
3x& - & 2y & - & 2z & = & 1 \\
\end{array}
\right.
$$
$$
\left\{
\begin{array}{rcrcrcrcr}
&  & y & - & 3z & +& 4v&= & 0 \\
x&  &  & - & 2z & &&= & 0 \\
3x& + & 2y &  &  & - & 5v&= & 2 \\
4x&  &  & - & 5z & &&= & 0 \\
\end{array}
\right.
$$
\texttt{Wyrażenie \#4}
$$
\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1 \\
\end{array}
\right]
\cdot 
\left[
\begin{array}{ccc}
1 & 2 & 3 \\
3 & 1 & 2 \\
5 & 1 & 3 \\
\end{array}
\right]
$$
$$
\left[
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
\end{array}
\right]
\cdot 
\left[
\begin{array}{cc}
11 & -2  \\
6 & -14 \\
-21& 30 \\
\end{array}
\right]
$$
$$
\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 1 \\
\end{array}
\right]
\cdot 
\left[
\begin{array}{ccc}
1 & 1 & 3 \\
2 & 1 &4  \\
1 & 3 & 0 \\
\end{array}
\right]
$$
\texttt{Wyrażenie \#5}
$$
\left\{
\begin{array}{rcrcrcrcr}
x&+  & 2y & + & 3z & +& t&= & 1 \\
2x& + & 4y & - & z & +&2t&= & 2 \\
3x& + & 6y & + & 10z & + & 3t&= & 3 \\
x& + & y & + & z & +&t&= & 0 \\
\end{array}
\right.
$$
$$
\left\{
\begin{array}{rcrcrcrcrcr}
x&-  & y & + & z &  - &2s&+& t&= & 0 \\
3x& + & 4y & - & z & +&s&+&3t&= & 1 \\
x& - & 8y & + & 5z & - &9s&+& t&= & -1 \\
\end{array}
\right.
$$
\texttt{Wyrażenie \#6}
$$
\left|
\begin{array}{rr}
-3 & 2 \\
8 & -5 
\end{array}
\right|
$$
$$
\left|
\begin{array}{rr}
\sin \alpha & \cos \alpha \\
\sin \beta & \cos \beta 
\end{array}
\right|
$$
$$
\left|
\begin{array}{ccc}
1& 1& 1 \\
1 & 2& 3\\
1& 3 & 6 
\end{array}
\right|
$$
$$
\left|
\begin{array}{ccc}
1& i& 1+i \\
-i & 1& 0\\
1-i& 0 & 1 
\end{array}
\right|
$$
\texttt{Wyrażenie \#7}
$$
B=
\left[
\begin{array}{c|cc|ccc}
1& 0 & 0 & 1 & 1& 1\\
\hline 
0 & 2 & 2 & 1 & 2 & 3\\
0 & 2 & 2 & 4 & 5 & 6\\
\hline
0& 0 & 0 & 3 & 3& 1\\
0& 0 & 0 & 3 & 1& 3\\
0& 0 & 0 & 1 & 3& 3\\
\end{array}
\right]
$$
\texttt{Wyrażenie \#8}
$$
\int_1^{\infty} \frac{dx}{(x+2)^2}
$$
$$
\int_{-\infty}^0 \frac{dx}{x^2+4}
$$
$$
\int_{-\infty}^{\infty} x^2e^{-x^3} dx
$$
$$
\int_1^{\infty} \frac{dx}{\sqrt[3]{3x+5}}
$$
$$
\int_{-1}^0 \frac{dx}{\sqrt[5]{x^2}}
$$
$$
\int_2^3 \frac{dx}{x^2-3x} 
$$
\texttt{Wyrażenie \#9}
$$\log_{\sqrt{5}} 5\sqrt[3]{5}$$
$$\log_{\sqrt[3]{3}} 27$$
$$\log_2 8\sqrt{2}$$
$$\log_{\frac{1}{3}}81\sqrt{3}$$
$$3^{2+\log_3 4}$$
$$2^{5-\frac{1}{3}\log _2 27}$$
$$\sqrt{10^{2+\frac{1}{2}\log 16}}$$
\texttt{Wyrażenie \#10}
$$
\int \frac{x^2 \, dx}{\sqrt{4-x^2}}
$$
$$
\int \frac{x^3 \, dx}{\sqrt{25+x^2}}
$$
$$
\int \sqrt{x^2-36}\, dx
$$
$$
\int \sqrt{3+x^2}\, dx
$$
\texttt{Wyrażenie \#11}
$$
\lim_{n\to\infty} 
\left( 
\sqrt{n+6\sqrt{n}+1}-\sqrt{n}
\right)
$$
$$
\lim_{n\to\infty}
\frac
{1+\frac{1}{2}+\frac{1}{2^2}+\ldots+\frac{1}{2^n}}
{1+\frac{1}{3}+\frac{1}{3^2}+\ldots+\frac{1}{3^n}}
$$
\texttt{Wyrażenie \#12}
$$
d_n=\cos \frac{\pi}{2n}
$$
$$
e_n=\sqrt[n]{5^n+6^n}
$$
$$
f_n=\frac{n!(2n)!}{(3n)!}
$$
\texttt{Wyrażenie \#13}
$$
\lim_{n\to\infty} \left(1+\frac{6}{n}\right)^n
$$
$$
\lim_{n\to\infty} \left(\frac{n}{n+1}\right)^{n+1}
$$
$$
\lim_{n\to\infty} \left(\frac{n+3}{n}\right)^{n+3}
$$
$$
\lim_{n\to\infty} \left(1-\frac{2}{n}\right)^{-n}
$$
\texttt{Wyrażenie \#14}
$$
\sum_{n=1}^{\infty} (-1)^{n+1}(2n-1)
$$
$$
\sum_{n=1}^{\infty} \sin\frac{2\pi}{3^n}\cos\frac{4\pi}{3^n}
$$
$$
\sum_{n=2}^{\infty} (\left(\sqrt[n]{n} - \sqrt[n+1]{n+1} \right)
$$
\end{document}