Calculus Notes: Limits, Domain Validity, and Vertical Asymptotes

Motion Modeling and Approaching Values

• A physical scenario involving an object moving from point $A$ to point $B$ can be represented using a position function, where the distance traveled is denoted by $s$.

• Effective study methodology and routine workflow:

  • Notes from completed sections should be systematically reviewed daily.

  • Questions arising from previous notes must be addressed at the start of the following session before moving on to new material.

• Numerical analysis of values approaching a specific target point $a$:

  • If $a = 2$, values of $x$ that lie close to $a$ from the right include $x = 2.1$ and $x = 2.01$.

  • Evaluating function values at sequential points such as $x_1$ yields corresponding output values $f(x_1)$ to analyze behavioral trends near $a$.

One-Sided Limits and Numerical Evaluations

• Left-Hand Limit:

  • Defined as the value that a function $f(x)$ approaches as $x$ approaches $a$ strictly from the left side (from values smaller than $a$).

  • Denoted as $\lim_{x \to a^-} f(x)$.

• Right-Hand Limit:

  • Defined as the value that a function $f(x)$ approaches as $x$ approaches $a$ strictly from the right side (from values larger than $a$).

  • Denoted as $\lim_{x \to a^+} f(x)$.

• Concrete evaluation using test values:

  • Consider the rational function $f(x) = \frac{x^2 - 4}{x - 2}$.

  • Evaluating $f(x)$ at the left-hand test point $x = 1.5$ yields:     $\frac{1.5^2 - 4}{1.5 - 2}$

  • Simplifying this numerical substitution:     $\frac{2.25 - 4}{-0.5} = \frac{-1.75}{-0.5} = 3.5$

Limit Existence Versus Functional Definition

• Evaluation of limits for $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$:

  • Left-hand limit as $x$ approaches $2$: $\lim_{x \to 2^-} f(x) = 4$.

  • Right-hand limit as $x$ approaches $2$: $\lim_{x \to 2^+} f(x) = 4$.

  • Because both the left-hand limit and right-hand limit approach the identical value $4$, the overall limit exists:     $\lim_{x \to 2} f(x) = 4$

• Functional value versus limit value:

  • Evaluating the function directly at $x = 2$ yields $\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$, which is undefined.

  • The function is not defined at $x = 2$, resulting in a hole (removable discontinuity) in the graph at $(2, 4)$.

  • Key Rule: A function does not need to be defined at $x = a$ for the limit as $x \to a$ to exist.

• Fundamental criteria for general limit existence:

  • The overall limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal:     $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

  • If the left-hand and right-hand limits are not equal, the overall limit Does Not Exist (DNE).

Domain Restrictions and Mathematical Function Definitions

• Formal Definition of a Function:

  • A mathematical relation is a valid function if and only if every element in the domain maps uniquely to exactly one element in the codomain.

• Violation of function criteria:

  • If an element $r$ in the domain maps to two or more distinct elements in the codomain, the mapping is non-unique.

  • Under non-unique mapping conditions, the relation fails to be a function, making evaluations of limits or functional values invalid.

• Notation flexibility:

  • Standard variable symbols (such as $a$) can be interchanged with other letters (such as $r$) to reinforce underlying conceptual definitions independent of specific variable names.

Infinite Limits and Rigorous Definition of Asymptotes

• Infinite Limits Overview:

  • Infinite limits analyze scenarios where function outputs grow without bound (towards positive or negative infinity) as $x$ approaches a point a$.\n- Domain exclusions and nature of infinity:\n - Consider a function where approaching x = 3 causes the output to grow infinitely.\n - Since infinity (\infty$) is a mathematical concept rather than a real number,$f(3) cannot map to a real number in the codomain.\n - To maintain a valid mathematical domain, x = 3 must be explicitly excluded from the domain of the function.\n - If a limit approaches infinity, it is stated to "exist as infinite" (or diverge to infinity), acknowledging that it does not equal a finite real value.\n- Rigorous Calculus Definition of Vertical Asymptotes:\n - High school definition: A visual description where a graph approaches a vertical line arbitrarily closely without touching it.\n - Rigorous limit-based definition: The vertical line x = a$is defined as a vertical asymptote of the function$f(x) if at least one of the following limit conditions is satisfied:\n    \lim_{x \to a^-} f(x) = +\infty\n    \lim_{x \to a^-} f(x) = -\infty\n    \lim_{x \to a^+} f(x) = +\infty\n    \lim_{x \to a^+} f(x) = -\infty\n    \lim_{x \to a} f(x) = +\infty\n    \lim_{x \to a} f(x) = -\infty\n - The notation \pm \infty$$ indicates that the function output approaches positive infinity on one side and negative infinity on the other side (or the same on both).

  • Without the formal mathematical concept of limits, it is impossible to rigorously define vertical or horizontal asymptotes.

Visual Intuition Versus Mathematical Rigor

• Descriptive observation versus formal definition:

  • Observing what a mathematical concept looks like (e.g., seeing a curve approach a line) is analogous to observing a physical object or structure.

  • Being able to visualize or interact with an object does not constitute a formal mathematical definition.

  • High school intuitive descriptions explain visual appearance, whereas calculus provides the formal framework (via limits) to define the underlying phenomenon precisely.