AP Calculus AB: Limits Graphically and Foundations

Course Information and Administrative Rules

• Course Overview:

  • The course covers material for AP Calculus AB.
  • The course does not cover AP Calculus BC material.

• Trademark and Copyright Notice:

  • AP stands for Advanced Placement, which is a copyrighted trademark term owned by the College Board.
  • College Board has no affiliation, endorsement, or involvement with the flip math website or these lessons.

• Teaching Staff:

  • The primary instructors for the course are mr. bean and mr. Bress.
  • mr. bean teaches Unit 1 and Unit 2.
  • mr. Bress takes over instruction starting in Unit 3.
  • Guest cameo visits may occur from mr. Sullivan and mr. Kelly.

• Calculator Policy:

  • Default assumption: Calculators are not to be used on lessons or practice problems unless specifically stated.
  • Calculator usage will be explicitly designated with a calculator symbol or specific problem instructions.
  • Certain lessons are explicitly centered around calculator usage.
  • Calculators are allowed on a designated portion of the AP exam, but non-calculator proficiency is vital due to the substantial portion of testing where calculators are forbidden.

• Lesson Structure:

  • Lessons are intentionally paced to be concise with minimal non-instructional commentary.

Fundamental Concept of Limits

• Definition of a Limit:

  • A limit is a $y$-value that a function's graph approaches from both the left side and the right side of a given $x$-value.

• Conceptual Approach vs. Formal Definition:

  • Informal / Graphical Approach: Tracking the trajectory of the graph from both the left and right sides of a given $x$-value to identify the target $y$-value.
  • Formal Mathematical Definition: Formal definitions (favored by mr. Kelly) are essential for advanced mathematical study, but graphical evaluation relies primarily on inspecting left-hand and right-hand trajectories.

Evaluating Functions versus Limits Graphically

• Scenario 1: Continuous Point

  • Graph characteristics: A solid filled point exists at $(-3, 4)$.
  • Function value: $f(-3) = 4$
  • Limit evaluation: $\lim_{x \rightarrow -3} f(x) = 4$

• Scenario 2: Removable Discontinuity (Hole without Displaced Point)

  • Graph characteristics: An open circle exists at $(-3, 4)$ with no filled dot present at $x = -3$
  • Function value: $f(-3)$ is undefined (or Does Not Exist / DNE).
  • Limit evaluation: $\lim_{x \rightarrow -3} f(x) = 4$
  • Principle: A limit measures where the graph is headed as $x$ approaches $-3$, regardless of whether a point exists at that exact coordinate.

• Scenario 3: Removable Discontinuity (Hole with Displaced Point)

  • Graph characteristics: An open circle exists at $(-3, 4)$, and a solid filled point exists at $(-3, 1)$.
  • Function value: $f(-3) = 1$
  • Limit evaluation: $\lim_{x \rightarrow -3} f(x) = 4$
  • Principle: The value of the limit is determined by the curve's approach from both sides, not by the location of the solid point.

• Scenario 4: Jump Discontinuity (Scenario A)

  • Graph characteristics: A solid filled point exists at $(-3, 1)$. As $x$ approaches $-3$, the left side of the graph goes to a higher $y$-value while the right side approaches a lower $y$-value.
  • Function value: $f(-3) = 1$
  • Limit evaluation: $\lim_{x \rightarrow -3} f(x) = \text{DNE}$ (Does Not Exist).
  • Principle: If the left side and right side of the graph approach different $y$-values as $x$ approaches $-3$, no general limit exists.

• Scenario 5: Jump Discontinuity (Scenario B)

  • Graph characteristics: A solid point exists at $(-3, 2)$. The left side approaches a $y$-value of $4$, and the right side approaches a $y$-value of $1$
  • Function value: $f(-3) = 2$
  • Limit evaluation: $\lim_{x \rightarrow -3} f(x) = \text{DNE}$

One-Sided Limits

• Definition:

  • A one-sided limit is the $y$-value that a function approaches as $x$ approaches a target value from strictly the left side or strictly the right side.

• Notation and Symbols:

  • Left-sided limit: $\lim_{x \rightarrow c^-} f(x)$ where the superscript minus sign ($^-$-) indicates approaching $c$ from values less than $c$ (from the left).
  • Right-sided limit: $\lim_{x \rightarrow c^+} f(x)$ where the superscript plus sign ($^+$) indicates approaching$c$from values greater than$c (from the right).\n\n* Example 2 Analysis:\n * Left-sided limit at x = 3$:$\lim_{x \rightarrow 3^-} f(x) = -1\n * Right-sided limit at x = 3$:$\lim_{x \rightarrow 3^+} f(x) = 2\n * Principle: An open circle does not prevent a one-sided limit from existing because the limit only evaluates the target value being approached.\n\n# Comprehensive Practice Problems (Example 3)\n\n* Evaluating Limits and Functions from Graph:\n * Part a: \lim_{x \rightarrow -2^-} f(x) = 1\n * Part b: \lim_{x \rightarrow -2^+} f(x) = 2\n * Part c: \lim_{x \rightarrow -2} f(x) = \text{DNE}$(because the left-hand limit$1$does not equal the right-hand limit$2)\n * Part d: \lim_{x \rightarrow 1} f(x) = 1$(both sides converge to$y = 1)\n * Part e: \lim_{x \rightarrow 0} f(x) = -2\n * Part f: \lim_{x \rightarrow 3^-} f(x) = 5\n * Part g: \lim_{x \rightarrow -1} f(x) = -3\n * Part h: \lim_{x \rightarrow -3} f(x) = 0\n * Part i: f(-2) = 1 (evaluated at the solid filled dot)\n * Part j: f(1) = -2 (evaluated at the solid filled dot)\n\n# Conditions for Non-Existence of Limits\n\n* Condition 1: Differing Left-Hand and Right-Hand Limits\n * The left-sided limit \lim_{x \rightarrow c^-} f(x)$and the right-sided limit$\lim_{x \rightarrow c^+} f(x)$approach two different$y-values.\n\n* Condition 2: Unbounded Behavior\n * As x$approaches a target value, the graph increases or decreases without bound toward$\infty$or$-\infty (such as at a vertical asymptote).\n * Note: Infinity is not a real number. While future notation may express such limits as \lim_{x \rightarrow c} f(x) = \infty$, technically the limit does not exist as a finite numerical$y-value.\n\n* Condition 3: Oscillating Behavior\n * As x$approaches a target value, the function infinitely oscillates back and forth between two fixed numbers without settling on a single$y-value.\n * This behavior typically occurs in trigonometric functions such as \sin\left(\frac{1}{x}\right)$near$x = 0.\n\n# Graphing Functions Given Specific Limit and Function Conditions\n\n* Problem Requirements for Graphing Function G(x):\n * Condition 1: G(3) = -1\n * Action: Plot a solid point at coordinate (3, -1).\n * Condition 2: \lim_{x \rightarrow 3} G(x) = 4\n * Action: Place an open circle at (3, 4) and draw curves approaching this point from both the left and right sides.\n * Condition 3: \lim_{x \rightarrow -2^+} G(x) = 1\n * Action: Draw the graph approaching a y$-value of$1$as$x$approaches$-2 from the right side.\n * Condition 4: G(x)$is increasing on the interval$(-2, 3)\n * Action: Ensure the graph sloped continuously upward from (-2, 1)$to$(3, 4).\n * Condition 5: \lim_{x \rightarrow -2^-} G(x) > \lim_{x \rightarrow -2^+} G(x)\n * Action: Draw the portion of the graph to the left of x = -2$such that as it approaches$x = -2$from the left, its$y$-value is strictly greater than$1\n\n* Verification and Quality Control Steps:\n * Re-check each listed condition against the drawn graph to verify compliance.\n * Perform the Vertical Line Test across the entire domain: A vertical line must cross the drawn graph at most once at any given x-value. Drawing extra overlapping pieces that fail the vertical line test will result in loss of credit.\n\n# Conceptual True/False Analysis\n\n* Evaluation of the Statement: "\lim_{x \rightarrow 2} f(x) = \text{DNE}"\n * Determination: False.\n * Explanation: As x$approaches$2$from both the left and right sides, the function approaches the same$y$-value of$2$. Therefore, the limit exists and equals$2$$.