Quantum Partition Function

The Quantum Partition Function

• Considers a system of $N$ identical particles within a volume $V$, interacting with a heat reservoir at temperature $T$.
• Energy levels are quantized in quantum mechanics, represented as $\epsilon_1, \epsilon_2, \epsilon_3,...$
• Classical mechanics uses a continuum of energy levels.
• The partition function is $Z = \sum_n e^{-\beta E_n}$, where $n$ labels the quantum states of the entire $N$-body system and $E_n$ is the energy of the $n$-th state, with $\beta = \frac{1}{k_B T}$.
• Occupation number $n_i$ shows the quantity of particles in a single-particle state $i$.
• The constraints are: $\sum_i n_i = N$ and $\sum_i n_i \epsilon_i = E_n$.
• The partition function can also be written as $Z = \sum_{{n_i}} e^{-\beta \sum_i n_i \epsilon_i}$, where ${n_i}$ denotes the possible occupation numbers.

Distinguishable Particles

• Each particle is uniquely identifiable.
• Found in solids where atoms are fixed to lattice locations.
• Partition function for $N$ atoms is $Z = \prod_i \frac{1}{1-e^{-\beta\epsilon_i}}$.

Indistinguishable Particles

• Particles are identical and cannot be distinguished.
• Common in gases where atoms move freely.
• Fermions obey the Pauli exclusion principle, with occupation numbers $n_i = 0, 1$.
• Bosons do not obey the Pauli exclusion principle, with occupation numbers $n_i = 0, 1, 2, 3,... \infty$.

Fermions

• The partition function is $Z = \prod_i (1 + e^{-\beta\epsilon_i})$.
• Pauli exclusion principle limits each state to either 0 or 1 particle.

Bosons

• The partition function is $Z = \sum_{n_1, n_2,...} e^{-\beta(n_1\epsilon_1 + n_2\epsilon_2 +...)}$.
• Occupation numbers range from 0 to infinity.
• Constraints $\sum_i n_i = N$ make calculation challenging.

Algorithmic Game Theory - Summer Term 2023

Key People

• Lecturers: Prof. Dr. Alexander Bilz and Dr. Sven Seuken

Logistics

• Lectures are held every Tuesday from 10:15 to 11:45 in Room 00-014, Building 079.
• Exercise sessions are every Tuesday from 12:15 to 13:45 in Room 00-014, Building 079.
• The course website is https://home.mathematik.uni-freiburg.de/bilz/teaching/ss2023/agt/.
• Exercise solutions can be submitted in groups up to 3 via email to [email protected] or [email protected].
• Bonus points for the final exam are calculated as: (Achieved points in exercises) / (Total points in exercises).

Course Content

• Topics include: introduction to game theory, auctions, mechanism design, congestion games, price of anarchy, and social choice.

Core Reading List

• Key texts include: Algorithmic Game Theory by Nisan, Roughgarden, Tardos, and Vazirani; Essentials of Game Theory: A Concise Multidisciplinary Introduction by Leyton-Brown and Shoham; Networks, Crowds, and Markets by Easly and Kleinberg; An Introduction to Game Theory by Osborne; and Game Theory by Maschler, Solan, and Zamir.

Lecture 14: Floating Point

Objectives

• Focuses on introducing the IEEE floating point standard and its implications for programming.

Motivations

• Explores the representation of real numbers, including scientific notation for very large or small numbers like $6.022 * 10^{23}$.

Representation

• Aims to represent very large numbers, very small numbers and fractions.
• Fixed point uses bits for both integer and fraction parts, but it is rarely used in modern computers.
• Floating point uses scientific notation with $(-1)^S * Significand * Base^{Exponent}$.

IEEE Floating Point Standard 754

• Single precision uses 32 bits (float in C), with 1 sign bit, 8 exponent bits, and 23 significand bits.
• Double precision uses 64 bits (double in C), with 1 sign bit, 11 exponent bits, and 52 significand bits.

Core Formula

• Expressed as $(-1)^S * (1 + Significand) * 2^{(Exponent - Bias)}$.
• The significand, also called the mantissa, is less than 1.
• The exponent is also known as the characteristic.
• Bias is subtracted to achieve a signed exponent, with a value of 127 for single precision and 1023 for double precision.

Special Values

• When the exponent equals 0, it represents zero if the significand is zero or can represent very small numbers known as denormalized numbers.
• Setting exponent to all ones designates infinity (if the significand is zero) and NaN (Not a Number) if non-zero, from operations such as division by zero.

Denormalized Numbers

• Numbers follow $(-1)^S * (0 + Significand) * 2^{(1 - Bias)}$.
• The exponent is zero while the significand is non-zero.
• These numbers fill the gap between 0 and the smallest normal number, but offer less precision.

Implications for Programming

• Floating point arithmetic is not exact and thus can lead to rounding errors.
• Use a tolerance when comparing floating point numbers to avoid issues with ==.
• Express this using something like if (fabs(a - b) < tolerance).

Considerations

• Floating point arithmetic is not associative, represented as $(a + b) + c != a + (b + c)$.
• Compilers may reorder floating point operations unless -ffloat-store command is employed to prevent it.
• Processors might use more precision during calculations, causing unexpected outcomes.

Static CMOS Inverter: DC Characteristics

Parameters

• $V_{OL}$: Output Low Voltage
• $V_{OH}$: Output High Voltage
• $V_{IL}$: Input Low Voltage
• $V_{IH}$: Input High Voltage
• $V_M$: Inverter Threshold Voltage

Voltage Transfer Characteristic

• Divided into five regions.
• Region A ($V_{in} \le V_{OL}$): NMOS OFF, PMOS ON, $V_{out} = V_{DD}$.
• Region B ($V_{OL} < V_{in} < V_M$): NMOS ON(Saturation), PMOS ON(Linear), $V_{out}$ decreases as $V_{in}$ increases.
• Region C ($V_{in} = V_M$): NMOS ON(Saturation), PMOS ON(Saturation), $V_{out} = V_{in} = V_M$.
• Region D ($V_M < V_{in} < V_{OH}$): NMOS ON(Linear), PMOS ON(Saturation), $V_{out}$ falls sharply as $V_{in}$ increases.
• Region E ($V_{in} \ge V_{OH}$): NMOS ON, PMOS OFF, $V_{out} = V_{SS}$.

Conditions

• $V_{OL}$ Condition: $V_{in} = V_{OH}$
• $V_{OH}$ Condition: $V_{in} = V_{OL}$

Equations

• Region B drain current equations:
  • $I_{DN} = \frac{k_n}{2}(V_{in} - V_{TN})^2(1 + \lambda_n V_{out})$
  • $I_{DP} = \frac{k_p}{2}[2(V_{in} - V_{DD} - V_{TP})(V_{out} - V_{DD}) - (V_{out} - V_{DD})^2](1 + \lambda_p V_{out})$
• where $I_{DN} = -I_{DP}$.
• Region D drain current Equations:
  • $I_{DN} = \frac{k_n}{2}[2(V_{in} - V_{TN})V_{out} - V_{out}^2](1 + \lambda_n V_{out})$
  • $I_{DP} = \frac{k_p}{2}(V_{in} - V_{DD} - V_{TP})^2(1 + \lambda_p V_{out})$
• where $I_{DN} = -I_{DP}$

Inverter Threshold Voltage

• $V_M$ when $V_{in} = V_{out} = V_M$.
  • $I_{DN} = \frac{k_n}{2}(V_M - V_{TN})^2(1 + \lambda_n V_M)$
  • $I_{DP} = \frac{k_p}{2}(V_M - V_{DD} - V_{TP})^2(1 + \lambda_p V_M)$
• If $I_{DN} = -I_{DP}$ and assuming $\lambda_n = \lambda_p = 0$, then:
  • $V_M = \frac{V_{DD} + V_{TP} + V_{TN}\sqrt{r}}{1 + \sqrt{r}}$, where $r = \frac{k_p}{k_n}$.
• To achieve $V_M = \frac{V_{DD}}{2}$, $r$ should be close to 1, implying $k_n = k_p$.

Noise Margins

• Noise Margin High ($NM_H$): $V_{OH} - V_{IH}$
• Noise Margin Low ($NM_L$): $V_{IL} - V_{OL}$

Noise Margin Definitions

• $V_{IL}$: the maximum input voltage interpreted as a LOW, where $\frac{dV_{out}}{dV_{in}} = -1$.
• $V_{IH}$: the minimum input voltage interpreted as a HIGH, where $\frac{dV_{out}}{dV_{in}} = -1$.

Calculation of $V_{IL}$ and $V_{IH}$

• $V_{IL}$ can be found using Region B equations:
  • $I_{DN} = \frac{k_n}{2}(V_{in} - V_{TN})^2$
  • $I_{DP} = \frac{k_p}{2}[2(V_{in} - V_{DD} - V_{TP})(V_{out} - V_{DD}) - (V_{out} - V_{DD})^2]$
• Differentiating $I_{DN} = -I_{DP}$ with respect to $V_{in}$, and setting $\frac{dV_{out}}{dV_{in}} = -1$.
• $V_{IH}$ can be found using Region D equations:
  • $I_{DN} = \frac{k_n}{2}[2(V_{in} - V_{TN})V_{out} - V_{out}^2]$
  • $I_{DP} = \frac{k_p}{2}(V_{in} - V_{DD} - V_{TP})^2$
• Differentiating $I_{DN} = -I_{DP}$ with respect to $V_{in}$, and setting $\frac{dV_{out}}{dV_{in}} = -1$.

Matrices

• Matrix: A rectangular array of numbers or symbols in rows and columns.
• General form:

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

• $a_{ij}$: Element in the $i$-th row and $j$-th column.
• $m$: Number of rows.
• $n$: Number of columns.
• Order of matrix $A: m \times n$.
• Example: $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ Order: $2 \times 3$, $a_{12} = 2$, $a_{23} = 6$

Types of Matrices

Square Matrix

• Has an equal number of rows and columns ($m = n$).

$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$

• Order: $2 \times 2$

Row Matrix

• A matrix with only one row ($m = 1$).

$A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

• Order: $1 \times 3$

Column Matrix

• A matrix with only one column ($n = 1$).

$A = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$

• Order: $3 \times 1$

Zero/Null Matrix

• A matrix in which all elements are zero.

$A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$

• Order: $2 \times 2$

Identity/Unit Matrix

• A square matrix with 1s on the main diagonal and 0s elsewhere. Denoted by $I$.

$I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

• Order: $2 \times 2$

Diagonal Matrix

• A square matrix in which all non-diagonal elements are zero.

$A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}$

• Order: $3 \times 3$

Triangular Matrix

• Upper Triangular Matrix: A square matrix in which all elements below the main diagonal are zero.

$A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}$

• Lower Triangular Matrix: A square matrix in which all elements above the main diagonal are zero.

$A = \begin{bmatrix} 1 & 0 & 0 \ 2 & 3 & 0 \ 4 & 5 & 6 \end{bmatrix}$

Matrix Operations

Addition

• Matrices can be added if they have the same dimensions.
• If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two matrices of the same order $m \times n$, their sum $C = A + B$ is a matrix of order $m \times n$, where each element $c_{ij} = a_{ij} + b_{ij}$.

Subtraction

• Similar to addition, matrices can be subtracted if they have the same dimensions.
• If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two matrices of the same order $m \times n$, their difference $C = A - B$ is a matrix of order $m \times n$, where each element $c_{ij} = a_{ij} - b_{ij}$.

Scalar Multiplication

• Multiplying a matrix by a constant.
• If $A = [a_{ij}]$ is a matrix of order $m \times n$ and $k$ is a scalar, then the scalar product $kA$ is a matrix of order $m \times n$, where each element is $ka_{ij}$.

Matrix Multiplication

• Two matrices $A$ and $B$ can be multiplied if the number of columns in $A$ is equal to the number of rows in $B$.
• If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then the product $C = AB$ is an $m \times p$ matrix. the element $c_{ij}$ of $C$ is obtained by multiplying the elements of the $i$-th row of $A$ with the corresponding elements of the $j$-th column of $B$ and summing them up.

$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$

Transpose of a Matrix

• The transpose of a matrix $A$, denoted as $A^T$ or $A'$, is obtained by interchanging its rows and columns.
• If $A$ is an $m \times n$ matrix, then $A^T$ is an $n \times m$ matrix.

Properties of Transpose

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(kA)^T = kA^T$, where $k$ is a scalar
• $(AB)^T = B^T A^T$

Determinant of a Matrix

• A scalar value computed from the elements of a square matrix.

Determinant of a $2 \times 2$ Matrix

• If $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, then $|A| = ad - bc$

Determinant of a $3 \times 3$ Matrix

• If $A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$, then $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

Properties of Determinants

• $|A^T| = |A|$
• If any two rows (or columns) are interchanged, the sign of the determinant changes.
• If any row (or column) is multiplied by a scalar $k$, the determinant is multiplied by $k$.
• If any two rows (or columns) are identical, the determinant is zero.
• If $A$ and $B$ are square matrices of the same order, then $|AB| = |A||B|$.

Adjoint of a Matrix

• For a square matrix $A$, the adjoint (or adjugate) of $A$, denoted as adj$(A)$, is the transpose of the matrix of cofactors of $A$.

Cofactor

• The cofactor $C_{ij}$ of an element $a_{ij}$ in a matrix $A$ is given by $C_{ij} = (-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$.

Adjoint Calculation

1. Find the matrix of cofactors of $A$.
2. Take the transpose of the matrix of cofactors to get adj$(A)$.

Inverse of a Matrix

• For a square matrix $A$, the inverse of $A$, denoted as $A^{-1}$, is a matrix such that $AA^{-1} = A^{-1}A = I$.

Formula for Inverse

• $A^{-1} = \frac{1}{|A|}\text{adj}(A)$
• Where $|A|$ is the determinant of $A$ and adj$(A)$ is the adjoint of $A$.

Condition for Invertibility

• A matrix $A$ is invertible if and only if its determinant $|A|$ is non-zero. If $|A| = 0$, the matrix is singular and does not have an inverse.

Properties of Inverse

• $(A^{-1})^{-1} = A$
• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^T)^{-1} = (A^{-1})^T$

Applications of Matrices

• Solving systems of linear equations: $AX = B$, so $X = A^{-1}B$
• Transformations in geometry, computer graphics, cryptography, and economics.

Planeten unseres Sonnensystems (Planets of our Solar System)

• The solar system consists of a central sun and the planets orbiting it.
• There are eight planets divided into inner rocky planets and outer gas planets.

Innere Gesteinsplaneten (Inner Rocky Planets)

• The inner planets are Mercury, Venus, Earth, and Mars.
• They are relatively small and primarily made of rock and metals.

Äußere Gasplaneten (Outer Gas Planets)

• The outer planets are Jupiter, Saturn, Uranus, and Neptune.
• They are much larger than the inner planets and primarily made of gasses like hydrogen and helium.

Weitere Objekte (Other Objects)

• In addition to the planets, there are many other objects in our solar system, such as moons, asteroids, and comets.

Guía del usuario de la aplicación móvil (Mobile App User Guide)

Primeros pasos (Get Started)

• Descarga e instalación (Download and Installation):
  • The mobile app is available for iOS and Android devices.
  • Download the app, visit App Store (iOS) or Google Play Store (Android) and search for "Nombre de la aplicación" (App Name).
• Crear una cuenta (Create an Account):
  • Open the application and click on the "Registrarse" (Sign Up) button.
  • Provide name, email address, and password. You must accept the Terms of Service and Privacy Policy.
• Iniciar sesión (Login):
  • Once they have created an account, enter your email address and password, then click on the "Iniciar sesión" (Log In) button.

Descripción general de la interfaz (Interface Overview)

The app interface is divided into five main tabs:

• Inicio (Home): Shows an overview of your account, the latest information.
• Explorar (Explore): Allows you to browse and search for interesting content.
• Crear (Create): Lets you create new content such as posts, stories, and videos.
• Notificaciones (Notifications): Displays notifications from your friends and other users.
• Perfil (Profile): Shows your profile information, including posts, stories, and videos.

Características (Features)

• Feeds: Displays the latest content from the people you follow.
• Explorar (Explore): Helps find new and interesting content from other users.
• Crear publicaciones (Create Posts): Can create posts and share them with friends and followers.
• Mensajería (Messaging): Send and receive messages from friends and other users.
• Notificaciones (Notifications): Stay up-to-date with the latest activity from your friends and other users.
• Configuración de perfil (Profile Settings): customize your profile by adding a profile photo, biography, and contact information.

Cómo utilizar la aplicación (How to use App)

• Cómo navegar por la aplicación (How to navigate the app):
  • Use the tabs at the bottom of the screen.
  • Each tab represents a new section of the app.
• Cómo buscar contenido (How to search for content):
  • Use the search bar at the top of the screen.
  • Enter keywords for the content you are looking for.
• Cómo crear una publicación (How to create a post):
  • Click the Create button at the bottom of the screen.
  • Enter the text of your post, add photos or videos, and choose privacy settings.
• Cómo enviar un mensaje (How to send messages):
  • Go to the profile of the person you want to message.
  • Click the Message button and enter your message, then click Send.
• Cómo administrar su perfil (How to manage your profile): Go to the Profile tab and click the Edit Profile button, where you can change your photo, bio, and contact information.

Solución de problemas (Troubleshooting)

• Problemas comunes (Common issues):
  • App not opening: make sure you have the most recent version of the app installed.
  • App crashing: try clearing cache and data for the app from the device settings. You can also try reinstalling the app.
  • Can't find content: Make sure you're using the proper search terms as you explore the application.
• Contacto con soporte técnico (Contact tech Support):
  • To contact support, go to the profile tab and click the Support button

Preguntas frecuentes (FAQ)

• ¿Cuál es el costo de usar la aplicación? (What is the cost of using the app?) - App is free to download
• ¿Cómo protejo mi privacidad? (How to protect my privacy?) - Adjust privacy settings under the settings to protect your privacy
• ¿Cómo informo un problema? (How to report a problem?) - To report a problem, go to the Profile Tab, and click the Support button

Lektion 3: Funktionen (Lesson 3: Functions)

3.1 Funktionsbegriff (Concept of a Function)

• Definition: A function from $A$ to $B$ is an assignment that assigns each element $x \in A$ to exactly one element $y \in B$.
  • $A$ is called the domain, $B$ is the range.
  • $x$ is called the independent variable, $y$ is the dependent variable.
  • $y$ is the function value of $x$, written $y = f(x)$.
  • The set of all function values is called the image of $f$: $Bild(f) = {f(x) | x \in A}$.
• Notation: $f: A \rightarrow B, x \mapsto f(x)$
• Beispiel (Example):
  • $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto x^2$
  • $f(x) = x^2$
  • $y = x^2$
• Graph einer Funktion (Graph of a Function):
  • Graph(f) = {(x, f(x)) | x \in A}
• Darstellung von Funktionen (Representation of Functions):
  • Funktionsterm (Function Term): $f(x) = x^2 + 1$
  • Graph
  • Wertetabelle (Value Table)
  • Pfeildiagramm (Arrow Diagram)
• Eigenschaften von Funktionen (Properties of Functions):
  • Injektiv (Injective): Each element of the image has exactly one pre-image.
  • Surjektiv (Surjective): The image is the entire range.
  • Bijektiv (Bijective): Injective and surjective.
• Umkehrfunktion (Inverse Function):
  • If $f: A \rightarrow B$ is bijective, then there exists the inverse function $f^{-1}: B \rightarrow A$.
• Beispiel (Example): $f(x) = 2x + 1$ ; $f^{-1}(y) = \frac{y - 1}{2}$
• Komposition von Funktionen (Composition of Functions):
  • Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be functions.
  • The composition of $f$ and $g$ is $g \circ f: A \rightarrow C$, defined as $(g \circ f)(x) = g(f(x))$.
• Beispiel (Example): $f(x) = x^2$, $g(x) = x + 1$; $(g \circ f)(x) = x^2 + 1$ and $(f \circ g)(x) = (x + 1)^2$

3.2 Elementare Funktionen (Elementary Functions)

• Polynomfunktionen (Polynomial Functions):
  • $f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$
  • $a_i$ are coefficients, $n$ is the degree of the polynomial function
• Rationale Funktionen (Rational Functions):
  • Functions are expressed as $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions.
• Exponentialfunktionen (Exponential Functions):
  • Functions are expressed as $f(x) = a^x$, where $a > 0$ and $a \neq 1$.
• Logarithmusfunktionen (Logarithmic Functions):
  • Functions are expressed as $f(x) = log_a(x)$, where $a > 0$ and $a \neq 1$.
• Trigonometrische Funktionen (Trigonometric Functions):
  • $f(x) = sin(x)$, $f(x) = cos(x)$, $f(x) = tan(x) = \frac{sin(x)}{cos(x)}$ Hyperbolische Funktionen (Hyperbolic Functions):
  • $f(x) = sinh(x) = \frac{e^x - e^{-x}}{2}$ $f(x) = cosh(x) = \frac{e^x + e^{-x}}{2}$ $f(x) = tanh(x) = \frac{sinh(x)}{cosh(x)}$
• Hyperbolische Funktionen (Hyperbolic Functions): - Functions include: $f(x) = sinh(x) = \frac{e^x - e^{-x}}{2}$, $f(x) = cosh(x) = \frac{e^x + e^{-x}}{2}$, $f(x) = tanh(x) = \frac{sinh(x)}{cosh(x)}$

3.3 Eigenschaften von Funktionen (Properties of Functions)

• Monotonie (Monotonicity):
  • Monoton steigend (Monotonically increasing): $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$
  • Monoton fallend (Monotonically decreasing): $x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)$
  • Streng monoton steigend (Strictly monotonically increasing): $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$
  • Streng monoton fallend (Strictly monotonically decreasing): $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$
• Symmetrie (Symmetry):
  • Achsensymmetrisch zur y-Achse (Axis-symmetric to the y-axis): $f(-x) = f(x)$ (gerade Funktion/even function)
  • Punktsymmetrisch zum Ursprung (Point-symmetric to the origin): $f(-x) = -f(x)$ (ungerade Funktion/odd function)
• Periodizität (Periodicity):
  • $f(x + T) = f(x)$ for all $x$ (T is the period)
• Stetigkeit (Continuity):
  • A function is continuous if small changes in $x$ lead to small changes in $f(x)$.
• Differenzierbarkeit (Differentiability):
  • A function is differentiable if it has a derivative.
• Grenzwerte (Limits):
  • $\lim_{x \to a} f(x) = L$ (The limit of $f(x)$ as $x$ approaches $a$ is $L$)
• Asymptoten (Asymptotes):
• Waagrechte Asymptote (Horizontal Asymptote): $\lim_{x \to \infty} f(x) = c$ (c is a constant)
  • Senkrechte Asymptote (Vertical Asymptote): $\lim_{x \to a} f(x) = \infty$
  • Schräge Asymptote (Oblique Asymptote): $f(x) = mx + b + g(x)$ with $\lim_{x \to \infty} g(x) = 0$