Advanced Mathematics I: Preliminaries and Limits

Preliminaries: Sets

• Definition of a Set:

  • A set is a well-defined collection of distinct objects, referred to as elements or members of the set.
  • Membership notation: If an element $x$ belongs to a set $A$, it is written as $x \in A$. If $x$ does not belong to $A$, it is written as $x \notin A$.

• Methods of Specifying Sets:

  • Roster (Listing) Method: Elements are explicitly listed within curly braces, e.g., $A = \{1, 2, 3, 4\}$.
  • Set-Builder Notation: Elements are defined by a characteristic property, e.g., $A = \{x \mid P(x)\}$, where $P(x)$ is a logical statement satisfied by all elements in A$.\n\n- Fundamental Sets of Numbers:\n - Natural Numbers: \mathbb{N} = {1, 2, 3, \dots}\n - Integers: \mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}\n - Rational Numbers: \mathbb{Q} = \left{\frac{p}{q} \mid p, q \in \mathbb{Z}, q eq 0\right}\n - Real Numbers: \mathbb{R}, representing all rational and irrational points along the continuous number line.\n - Complex Numbers: \mathbb{C} = {a + b i \mid a, b \in \mathbb{R}, i = \sqrt{-1}}\n\n- Subsets, Special Sets, and Set Relations:\n - Subset: A set A$is a subset of$B$($A \subseteq B$) if every element of$A$is also an element of$B.\n - Proper Subset: A$is a proper subset of$B$($A \subset B$) if$A \subseteq B$and$A eq B.\n - Empty Set: The set containing no elements, denoted by \emptyset or \{\}. The empty set is a subset of every set.\n - Universal Set: The total set of all possible objects under consideration in a given context, denoted by U.\n\n- Set Operations:\n - Union: A \cup B = {x \mid x \in A \text{ or } x \in B}\n - Intersection: A \cap B = {x \mid x \in A \text{ and } x \in B}\n - Set Difference (Relative Complement): A \setminus B = {x \mid x \in A \text{ and } x otin B}\n - Absolute Complement: A^c = U \setminus A = {x \in U \mid x otin A}\n\n- Interval Notation on Real Line \mathbb{R}:\n - Open Interval: (a, b) = {x \in \mathbb{R} \mid a < x < b}$
  • Closed Interval: [a, b] = \{x \in \mathbb{R} \mid a \le x \le b\}$\n - Half-Open / Half-Closed Intervals: [a, b) = {x \in \mathbb{R} \mid a \le x < b}$and$(a, b] = {x \in \mathbb{R} \mid a < x \le b}$
  • Infinite Intervals: $(a, \infty) = \{x \in \mathbb{R} \mid x > a\}$, $[a, \infty) = \{x \in \mathbb{R} \mid x \ge a\}$, $(-\infty, b) = \{x \in \mathbb{R} \mid x < b\}$, $(-\infty, b] = \{x \in \mathbb{R} \mid x \le b\}$, and $(-\infty, \infty) = \mathbb{R}$

• Algebraic Properties of Set Operations:

  • Commutative Laws: $A \cup B = B \cup A$ and $A \cap B = B \cap A$
  • Associative Laws: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$
  • Distributive Laws: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
  • De Morgan's Laws: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$

Preliminaries: The Rectangular Coordinate System

• Structure of the Cartesian Plane $\mathbb{R}^2$:

  • Constructed by two perpendicular real number lines intersecting at their zero points, called the origin $O(0, 0)$.
  • Horizontal axis: x-axis. Vertical axis: y-axis.
  • Any point $P$ in the plane is uniquely identified by an ordered pair of real numbers $(x, y)$, where $x$ is the directed horizontal distance (abscissa) and $y$ is the directed vertical distance (ordinate).

• Quadrants and Sign Conventions:

  • Quadrant I: $x > 0, y > 0$
  • Quadrant II: $x < 0, y > 0$
  • Quadrant III: $x < 0, y < 0$
  • Quadrant IV: $x > 0, y < 0$

• Distance Formula:

  • Derived from the Pythagorean Theorem, the distance $d$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ in the plane is given by:   $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula:

  • The midpoint $M(x_m, y_m)$ of the line segment connecting $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ divides the segment into two equal parts:   $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Preliminaries: The Straight Line

• Slope and Inclination Angle:

  • The angle of inclination $\theta$ of a line is the counterclockwise angle measured from the positive x-axis to the line, where $0 \le \theta < \pi$ radians ($0^\circ \le \theta < 180^\circ$).
  • The slope $m$ of a non-vertical line passing through two distinct points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is defined as:   $m = \tan(\theta) = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$
  • Vertical lines have an inclination angle of $\theta = \frac{\pi}{2}$ ($90^\circ$) and their slope is undefined. Horizontal lines have $\theta = 0$ and slope m = 0$.\n\n- Equations of Straight Lines:\n - Point-Slope Form: For a line passing through (x_1, y_1)$with slope$m:\n  y - y_1 = m(x - x_1)\n - Slope-Intercept Form: For a line with slope m$and y-intercept$b$(crossing the y-axis at$(0, b)):\n  y = m x + b\n - Two-Point Form: For a line passing through (x_1, y_1)$and$(x_2, y_2)$where$x_1 eq x_2$and$y_1 eq y_2:\n  \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\n - Intercept Form: For a line with x-intercept a eq 0$and y-intercept$b eq 0:\n  \frac{x}{a} + \frac{y}{b} = 1\n - General Linear Form: Any straight line in the Cartesian plane can be represented by:\n  A x + B y + C = 0\n  where A$and$B are not both zero.\n\n- Parallel and Perpendicular Lines:\n - Parallel Lines: Two non-vertical lines L_1$and$L_2$with slopes$m_1$and$m_2$are parallel ($L_1 \parallel L_2) if and only if their slopes are equal:\n  m_1 = m_2\n - Perpendicular Lines: Two non-vertical lines L_1$and$L_2$with slopes$m_1$and$m_2$are perpendicular ($L_1 \perp L_2$) if and only if the product of their slopes is$-1:\n  m_1 \times m_2 = -1 \quad \text{or} \quad m_2 = -\frac{1}{m_1}\n\n- Perpendicular Distance from a Point to a Line:\n - The shortest distance d$from a point$P_0(x_0, y_0)$to a line given by$A x + B y + C = 0 is calculated using:\n  d = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^2 + B^2}}\n\n# Preliminaries: Graphs of Equations\n\n- Definition of Graph of an Equation:\n - The graph of an equation in two variables x$and$y$consists of the set of all points$(x, y) in the coordinate plane whose coordinates satisfy the equation.\n\n- Intercept Analysis:\n - x-intercepts: Points where the graph intersects or touches the x-axis, determined by setting y = 0$and solving for$x$.
  • y-intercepts: Points where the graph intersects or touches the y-axis, determined by setting $x = 0$ and solving for y$.\n\n- Tests for Symmetry:\n - Symmetry with respect to the y-axis: Replacing x$with$-x$in the equation yields an equivalent equation. For a function$y = f(x)$, this corresponds to an even function where$f(-x) = f(x).\n - Symmetry with respect to the x-axis: Replacing y$with$-y in the equation yields an equivalent equation.\n - Symmetry with respect to the Origin: Replacing both x$with$-x$and$y$with$-y$yields an equivalent equation. For a function$y = f(x)$, this corresponds to an odd function where$f(-x) = -f(x).\n\n- Equations of Circles:\n - Standard Form: The equation of a circle with center (h, k)$and radius$r > 0 is:\n  (x - h)^2 + (y - k)^2 = r^2\n - General Form: Expanding the standard form yields:\n  x^2 + y^2 + D x + E y + F = 0\n  where D^2 + E^2 - 4 F > 0$. The center is at$\left(-\frac{D}{2}, -\frac{E}{2}\right)$and radius$r = \frac{1}{2}\sqrt{D^2 + E^2 - 4 F}.\n\n# Limits: Introduction and Conceptual Framework\n\n- Intuitive Concept of a Limit:\n - The limit of a function f(x)$as$x$approaches a fixed value$a$describes the value that$f(x)$approaches as$x$gets arbitrarily close to$a$without actually equaling$a$.
  • Expressed notationally as:   $\lim_{x \rightarrow a} f(x) = L$
  • The actual value of $f(a)$ is independent of $\lim_{x \rightarrow a} f(x)$. The function may be undefined at $x = a$, equal to $L$, or equal to a different value altogether.

• One-Sided Limits:

  • Left-Hand Limit: The value $L_1$ that $f(x)$ approaches as $x$ approaches $a$ strictly from values less than $a$ ($x < a$):   $\lim_{x \rightarrow a^-} f(x) = L_1$
  • Right-Hand Limit: The value $L_2$ that $f(x)$ approaches as $x$ approaches $a$ strictly from values greater than $a$ ($x > a$):   $\lim_{x \rightarrow a^+} f(x) = L_2$

• Necessary and Sufficient Condition for Limit Existence:

  • The overall limit $\lim_{x \rightarrow a} f(x)$ exists and equals $L$ if and only if both the left-hand limit and right-hand limit exist and are equal to $L$:   $\lim_{x \rightarrow a} f(x) = L \iff \lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = L$

Limits: Rigorous Definition

• The Formal $\epsilon$-$\delta$ Definition (Cauchy Definition):

  • Let $f(x)$ be a function defined on an open interval containing $a$, except possibly at $a$ itself. The statement $\lim_{x \rightarrow a} f(x) = L$ means that for every real number $\epsilon > 0$, there exists a corresponding real number $\delta > 0$ such that for all $x$:   $0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$

• Breakdown of Terms in the Definition:

  • $\epsilon$ (epsilon): Represents an arbitrary error tolerance bound for the output value $f(x)$ relative to the limit L$.\n - \delta$(delta): Represents a corresponding input distance tolerance around$a$.
  • $0 < |x - a|$: Specifies a deleted neighborhood of $a$, excluding $x = a$ from consideration.
  • $|f(x) - L| < \epsilon$: Guarantees that $f(x)$ stays strictly within the open interval $(L - \epsilon, L + \epsilon)$.

• Formal Definitions for One-Sided Limits:

  • Left-Hand Limit Definition: $\lim_{x \rightarrow a^-} f(x) = L$ means for every $\epsilon > 0$, there exists $\delta > 0$ such that if $a - \delta < x < a$, then |f(x) - L| < \epsilon$.\n - Right-Hand Limit Definition: \lim_{x \rightarrow a^+} f(x) = L$means for every$\epsilon > 0$, there exists$\delta > 0$such that if$a < x < a + \delta$, then$|f(x) - L| < \epsilon$.

Limits: Operations and Limit Laws

• Fundamental Limit Laws:

  • Assuming that $\lim_{x \rightarrow a} f(x) = L$ and $\lim_{x \rightarrow a} g(x) = M$ both exist:
  • Sum Law:     $\lim_{x \rightarrow a} [f(x) + g(x)] = L + M$
  • Difference Law:     $\lim_{x \rightarrow a} [f(x) - g(x)] = L - M$
  • Constant Multiple Law:     $\lim_{x \rightarrow a} [c \times f(x)] = c \times L \quad (c \in \mathbb{R})$
  • Product Law:     $\lim_{x \rightarrow a} [f(x) \times g(x)] = L \times M$
  • Quotient Law:     $\lim_{x \rightarrow a} \left[\frac{f(x)}{g(x)}\right] = \frac{L}{M} \quad (M \neq 0)$
  • Power Law:     $\lim_{x \rightarrow a} [f(x)]^n = L^n \quad (n \in \mathbb{N})$
  • Root Law:     $\lim_{x \rightarrow a} \sqrt[n]{f(x)} = \sqrt[n]{L}$     (where $L > 0$ if $n$ is even).

• The Squeeze Theorem (Sandwich Theorem):

  • If $g(x) \le f(x) \le h(x)$ for all $x$ in an open interval containing $a$ (except possibly at $a$ itself), and:   $\lim_{x \rightarrow a} g(x) = L \quad \text{and} \quad \lim_{x \rightarrow a} h(x) = L$   then:   $\lim_{x \rightarrow a} f(x) = L$

Limits: Limits at Infinity and Infinite Limits

• Limits at Infinity (Behavior as $x \rightarrow \pm\infty$):
  • Limit as $x \rightarrow \infty$: $\lim_{x \rightarrow \infty} f(x) = L$ means for every $\epsilon > 0$, there exists $M > 0$ such that for all $x > M$, |f(x) - L| < \epsilon$.\n - Limit as x \rightarrow -\infty$:$\lim_{x \rightarrow -\infty} f(x) = L$means for every$\epsilon > 0$, there exists$N < 0$such that for all$x < N$,$|f(x) - L| < \epsilon$.
  • Horizontal Asymptotes: The line $y = L$ is a horizontal asymptote of the graph of $y = f(x)$ if either $\lim_{x \rightarrow \infty} f(x) = L$ or \lim_{x \rightarrow -\infty} f(x) = L$.\n\n- Infinite Limits (Unbounded Function Growth):\n - Limit Equals Positive Infinity: \lim_{x \rightarrow a} f(x) = \infty$means for every positive constant$B > 0$, there exists a$\delta > 0$such that if$0 < |x - a| < \delta$, then$f(x) > B$.
  • Limit Equals Negative Infinity: $\lim_{x \rightarrow a} f(x) = -\infty$ means for every negative constant $B < 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then f(x) < B$.\n - Vertical Asymptotes: The line x = a$is a vertical asymptote of the graph of$y = f(x)$if at least one of the one-sided limits approaches$\infty$or$-\infty$as$x \rightarrow a^+$or$x \rightarrow a^-.\n\n# Limits: Infinitely Small Quantities (Infinitesimals)\n\n- Definition of an Infinitesimal:\n - A function or variable quantity \alpha(x)$is defined as an infinitely small quantity (or infinitesimal) as$x \rightarrow a$if its limit as$x \rightarrow a is equal to zero:\n  \lim_{x \rightarrow a} \alpha(x) = 0\n - An infinitesimal is a variable quantity approaching zero, not a static fixed non-zero value.\n\n- Relationship Between Limits and Infinitesimals:\n - A function f(x)$has a limit$L$as$x \rightarrow a$if and only if$f(x)$can be expressed as the sum of$L$and an infinitesimal$\alpha(x)$as$x \rightarrow a:\n  \lim_{x \rightarrow a} f(x) = L \iff f(x) = L + \alpha(x) \quad \text{where} \quad \lim_{x \rightarrow a} \alpha(x) = 0\n\n- Infinitely Large Quantities and Reciprocal Duality:\n - A function \beta(x)$is an infinitely large quantity as$x \rightarrow a$if$\lim_{x \rightarrow a} |\beta(x)| = \infty\n - Reciprocal Duality Theorem: If \alpha(x)$is an infinitesimal with$\alpha(x) eq 0$as$x \rightarrow a$, then its reciprocal$\frac{1}{\alpha(x)}$is an infinitely large quantity. Conversely, if$\beta(x)$is an infinitely large quantity, then$\frac{1}{\beta(x)} is an infinitesimal.\n\n- Comparison and Hierarchy of Infinitesimals:\n - Let \alpha(x)$and$\beta(x)$be two infinitesimals as$x \rightarrow a$with$\beta(x) eq 0$near$a:\n - Higher Order Infinitesimal: If \lim_{x \rightarrow a} \frac{\alpha(x)}{\beta(x)} = 0$, then$\alpha(x)$is an infinitesimal of higher order than$\beta(x)$, written as$\alpha(x) = o(\beta(x)).\n - Lower Order Infinitesimal: If \lim_{x \rightarrow a} \frac{\alpha(x)}{\beta(x)} = \infty$, then$\alpha(x)$is an infinitesimal of lower order than$\beta(x).\n - Same Order Infinitesimal: If \lim_{x \rightarrow a} \frac{\alpha(x)}{\beta(x)} = C$where$C eq 0$is a constant, then$\alpha(x)$and$\beta(x) are infinitesimals of the same order.\n - Infinitesimal of Order k$: If$\lim_{x \rightarrow a} \frac{\alpha(x)}{[\beta(x)]^k} = C$where$C eq 0$and$k > 0$, then$\alpha(x)$is an infinitesimal of order$k$relative to$\beta(x).\n - Equivalent Infinitesimals: If \lim_{x \rightarrow a} \frac{\alpha(x)}{\beta(x)} = 1$, then$\alpha(x)$and$\beta(x)$are equivalent infinitesimals as$x \rightarrow a$, written as$\alpha(x) \sim \beta(x).\n\n- Standard Equivalent Infinitesimals as x \rightarrow 0:\n - \sin(x) \sim x\n - \tan(x) \sim x\n - \arcsin(x) \sim x\n - \arctan(x) \sim x\n - 1 - \cos(x) \sim \frac{1}{2} x^2\n - e^x - 1 \sim x\n - \ln(1 + x) \sim x\n - (1 + x)^a - 1 \sim a x\n\n- Equivalent Infinitesimal Replacement Principle:\n - If \alpha(x) \sim \alpha'(x)$and$\beta(x) \sim \beta'(x)$as$x \rightarrow a$, and$\lim_{x \rightarrow a} \frac{\alpha'(x)}{\beta'(x)} exists, then:\n  \lim_{x \rightarrow a} \frac{\alpha(x)}{\beta(x)} = \lim_{x \rightarrow a} \frac{\alpha'(x)}{\beta'(x)}$$
  • Equivalent infinitesimals can be substituted into products and quotients during limit evaluations to simplify complex expressions.