Comprehensive Study Notes on Limits of Functions and Laboratory Workflow

Today's Agenda and Course Plan

• Section 2.2 Completion: Continuation and finalization of the discussion on the limit of a function.
• Class Break: A scheduled short break between the lecture portion and the laboratory session.
• Lab 1 Implementation: Hands-on work on Lab 1 during the final segment of class, including demonstration of workflow, explanation of analytical approaches, and walking around for individual assistance.

Informal Behavior of Functions and Non-Existent Limits

• Unbounded Growth Example: Consider the function $h(x) = \frac{1}{(x-3)^2}$.
  • As the input $x$ approaches $2$ (or $3$ depending on graphical center), the output values continuously grow without bound.
  • As $x$ approaches from the left, $y$-values grow infinitely large.
  • As $x$ approaches from the right, choosing inputs very close to the target value yields massive positive $y$-values.
• Mathematical Interpretation of Infinite Behavior:
  • Because the outputs keep increasing endlessly, they do not approach a single, finite real number.
  • Therefore, the limit does not exist (written as $\text{DNE}$).
  • Colloquially, this behavior is denoted as approaching $+\infty$; however, infinity is not a real number, making "does not exist" the formally correct statement.

Precise Mathematical Definition of a Limit

• Formal Definition:
  • Let $f$ be a function defined on an open interval containing $a$, with the possible exception of $a$ itself.
  • We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$.
• Formal Mathematical Notation:
  • Written concisely as: $\lim_{x \to a} f(x) = L$
  • Read aloud as: "The limit as $x$ approaches $a$ of $f(x)$ is equal to $L$."
• Crucial Conceptual Rules:
  • Exclusion of $a$: When investigating $\lim_{x \to a} f(x)$, inputs where $x = a$ are explicitly excluded. Limits evaluate the behavior of $f(x)$ strictly as $x$ gets closer and closer to $a$.
  • Irrelevance of $f(a)$: The exact value of $f(a)$ has zero bearing on the existence or value of $\lim_{x \to a} f(x)$. The function $f(a)$ can be undefined, equal to $L$, or equal to an entirely different number.

Fundamental Conceptual Examples of Limits

• Example 1: Undefined Function Value with a Valid Limit

  • Consider a function $f(x)$ whose graph contains a removable discontinuity (a hole) at $x = 2$ with no defined point above or below it.
  • Evaluations:
  • Substituting $x = 2$ yields division by zero in the denominator ($\frac{0}{0}$ form), making $f(2)$ completely undefined.
  • As $x$ approaches $2$ from the left, $f(x)$ approaches $4$.
  • As $x$ approaches $2$ from the right, choosing a point extremely close to $2$ yields a $y$-value extremely close to 4$.\n - **Conclusion**:\n - f(2) = \text{undefined}\n - \lim_{x \to 2} f(x) = 4\n\n- **Example 2: Displaced Function Value (Piecewise Point)**\n - Consider a function g(x)$whose graph follows a continuous curve toward$y = 4$at$x = -1$, but contains a hole at$(-1, 4)$and a solid dot located at$(-1, 5).\n - **Evaluations**:\n - Evaluating the function directly at x = -1$gives$g(-1) = 5$.
  • Choosing values of $x$ arbitrarily close to $-1$ from either the left or the right yields outputs that get closer and closer to 4$.\n - **Conclusion**:\n - g(-1) = 5\n - \lim_{x \to -1} g(x) = 4\n - **Takeaway**: Ignore the isolated point g(a) and focus exclusively on the trend of the surrounding graph.\n\n# Estimating Limits Using Numerical Tables\n\n- **Methodology**: When algebraic techniques are not yet established, limits can be estimated by evaluating the function at inputs progressively closer to the target value from both sides.\n\n- **Textbook Example: f(x) = \frac{\sin(x)}{x}$as$x \to 0**\n - Evaluating from the right (x > 0):\n - x = 0.1 \implies f(0.1) \approx 0.9983\n - x = 0.01 \implies f(0.01) \approx 0.99998\n - x = 0.001 \implies f(0.001) \approx 0.9999998\n - Evaluating from the left (x < 0):\n - x = -0.1 \implies f(-0.1) \approx 0.9983\n - x = -0.01 \implies f(-0.01) \approx 0.99998\n - x = -0.001 \implies f(-0.001) \approx 0.9999998\n - **Conclusion**: As x \to 0$,$f(x) \to 1$. Therefore,$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, even though$f(0) is undefined.\n\n- **Textbook Example 2.4 / 2.5: Limit as x \to 4**\n - Evaluating from the right (x > 4):\n - x = 4.1 \implies f(x) \approx 0.248\n - x = 4.01 \implies f(x) \approx 0.2498\n - Evaluating from the left (x < 4):\n - x = 3.9 \implies f(x) \approx 0.2516\n - x = 3.99 \implies f(x) \approx 0.2501\n - **Conclusion**: The function values approach 0.25$(or as an exact fraction,$\frac{1}{4}$). Thus,$\lim_{x \to 4} f(x) = 0.25 = \frac{1}{4}.\n\n# One-Sided Limits: Left-Hand and Right-Hand Definitions\n\n- **Motivation**: Functions may exhibit completely different behaviors on either side of a point, or may only be defined on one side (e.g., \sqrt{x}$or$\ln(x)$near$x = 0).\n\n- **Left-Hand Limit**:\n - Definition: The value L$that$f(x)$approaches as$x$approaches$a$strictly for values less than$a$($x < a).\n - Notation: \lim_{x \to a^-} f(x) = L\n - The superscript minus sign (^-$n) denotes approaching from the left (negative direction relative to $a$).

• Right-Hand Limit:

  • Definition: The value $L$ that $f(x)$ approaches as $x$ approaches $a$ strictly for values greater than $a$ ($x > a$).
  • Notation: $\lim_{x \to a^+} f(x) = L$
  • The superscript plus sign ($^+$) denotes approaching from the right (positive direction relative to$a).\n\n# Piecewise Function Example with One-Sided Limits\n\n- **Function Definition**:\n - Let f(x) be defined as:\n    f(x) = \begin{cases} x + 1 & \text{if } x < 2 \ x^2 - 4 & \text{if } x \ge 2 \end{cases}\n\n- **Direct Evaluations**:\n - f(3)$: Since $3 \ge 2$, evaluate using $x^2 - 4 \implies 3^2 - 4 = 5$.
  • f(1)$: Since 1 < 2$, evaluate using$x + 1 \implies 1 + 1 = 2\n\n- **One-Sided Limit Evaluations at x = 2**:\n - **Part A: Left-Hand Limit (\lim_{x \to 2^-} f(x))**:\n - For x < 2$, the active rule is$f(x) = x + 1$.
  • As $x$ approaches $2$ from the left, $f(x)$ approaches 2 + 1 = 3$.\n - \lim_{x \to 2^-} f(x) = 3\n - **Part B: Right-Hand Limit (\lim_{x \to 2^+} f(x))**:\n - For x > 2$, the active rule is$f(x) = x^2 - 4$.
  • As $x$ approaches $2$ from the right, $f(x)$ approaches 2^2 - 4 = 0$.\n - \lim_{x \to 2^+} f(x) = 0\n\n# Theorem on the Existence of Two-Sided Limits\n\n- **Theorem Statement**:\n - Let f(x)$be a function defined on an open interval around$a$(except possibly at$a$), and let$L be a real number.\n - The two-sided limit exists if and only if both one-sided limits exist and are equal to L:\n    \lim_{x \to a} f(x) = L \iff \left(\lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L\right)\n\n- **Consequences**:\n - If \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$, then the overall limit$\lim_{x \to a} f(x)$**does not exist** ($\text{DNE}).\n - In the piecewise example above, since \lim_{x \to 2^-} f(x) = 3$and$\lim_{x \to 2^+} f(x) = 0$,$\lim_{x \to 2} f(x) = \text{DNE}.\n\n# Non-Existence of Limits via Infinite Oscillation\n\n- **Function Analysis**: Consider f(x) = \sin\left(\frac{1}{x}\right)$as$x \to 0$.

• Numerical Table Behavior:

  • Testing values near zero yields erratic outputs:
  • $x = 0.1 \implies -0.5$
  • $x = 0.01 \implies 0.8$
  • $x = 0.001 \implies -0.3$

• Graphical Behavior:

  • As $x$ gets closer to $0$, $\frac{1}{x}$ approaches infinity rapidly, causing $\sin\left(\frac{1}{x}\right)$ to oscillate infinitely fast between $-1$ and 1$.\n - The function never settles down or stabilizes near any single fixed value L$.

• Conclusion: $\lim_{x \to 0} \sin\left(\frac{1}{x}\right) = \text{DNE}$.

Lab 1 Overview, Application Features, and Problem Walkthroughs

• General Setup and Requirements:

  • Access Lab 1 through SQN / Blackboard.
  • Ensure your full name is entered at the top of the lab report document.

• Problem 1 (Page 1):

  • Focuses on function $f_1(x)$.
  • Predict $f_1(1)$ by viewing the graph and table of values as $x$ approaches $1$ from left and right.

• Problem 2 (Page 2):

  • Uses variable Zoom Levels on function $f_1(x)$ and function $f_3(x)$.
  • Evaluate and make predictions across specified zoom thresholds: Zoom Level 3, Zoom Level 20, and Zoom Level 34.
  • Compare numerical table predictions against initial predictions in Problem 1.
  • Repeat the exact evaluation sequence for function $f_3(x)$.

• Problem 3 (Page 3):

  • Evaluates function $f_2(x)$.
  • Toggle UI controls to separate evaluations between "Left" and "Right" limits.
  • Compare left-hand and right-hand limits to state whether $\lim_{x \to a} f_2(x)$ exists.

• Problem 4 (Page 4):

  • Evaluates function $f_4(x)$ as x \to -1$.\n - Use higher zoom levels to determine \lim_{x \to -1} f_4(x).\n - Uncheck/unclick the "Hide" box in the user interface to reveal the hidden point value for f_4(-1)$$.

Lab 1 Submission Guidelines and Technical Policies

• Mandatory PDF Export Procedure (To Prevent Corruption):
  • DO NOT submit the raw fillable PDF file directly to Blackboard, as fillable form fields can suffer technical submission glitches.
  • Steps to Export:
  1. Complete all answers in the lab report document.
  2. Press Ctrl + P (or select Print).
  3. Change printer destination to "Print to PDF" or "Save as PDF".
  4. Save the file locally as a static PDF (e.g., GoXL_lab1_report.pdf).
  5. Drag and drop the static PDF file into the submission box on Blackboard and click Submit.
• Late Submission Policy:
  • Late lab submissions are strictly not accepted. The submission portal automatically locks upon the deadline.