Introduction to Limits and Numerical, Graphical, and Algebraic Evaluation Methods

Course Administration & Classroom Technologies

• iClicker and Location Sensitivity:

  • iClicker questions require logging in via a phone, laptop, or dedicated device.
  • Next week, iClicker attendance will become location-sensitive. Accessing the system will strictly require connecting to the UAV Wi-Fi network.
  • Discussion with surrounding peers is encouraged during iClicker questions to confirm answers prior to submission.
  • Students experiencing technical difficulties or arriving late can complete a physical yellow form prior to leaving class to adjust their score to $100\%$.

• Office Hours and Course Structure:

  • Virtual office hours take place on Zoom from 5:00 PM to 6:00 PM.
  • Additional assistance is available in the Math Learning Lab.
  • Standard course pace covers approximately one lesson per class period, totaling two lessons per week (Tuesdays and Thursdays).
  • Course notes function as a comprehensive reference textbook. Class periods focus on core conceptual understanding rather than completing every printed practice problem.

Calculator Setup and Numerical Evaluation Procedures

• Graphing Calculator Key Sequences (TI-84 / TI-89):
  • Defining Functions: Press Y= (top left corner). Input the target function using the variable key [X, T, \theta, n] located directly under MODE.
  • Unary vs. Binary Operations:
  • The negative key (-) represents a unary operation (takes no preceding operand).
  • The minus key (-) represents a binary operation (requires values before and after the operator).
  • Using the negative key in place of the minus key results in a syntax error.
  • Configuring Table Settings (TBLSET):
  • Press [2nd] then [WINDOW] (TBLSET).
  • Change the independent variable setting (Indpnt) from Auto to Ask to allow manual input entry.
  • Accessing the Table (TABLE):
  • Press [2nd] then [GRAPH] (TABLE).
  • Clear existing entries using the [DEL] key.
  • Enter desired $x$-inputs to evaluate corresponding $y$-outputs instantly.
  • Fraction Entry Shortcuts:
  • Method 1 (Newer Models): Press [MATH], arrow over to FRAC, and select Option 1.
  • Method 2 (Shortcut): Press [ALPHA] then [Y=] (or [X]).
  • Method 3 (Older Models/TI-89): Enclose numerators and denominators in explicit parentheses: (numerator)/(denominator).
  • Converting Fraction Outputs to Decimals:
  • In the [Y=] menu or main screen, press [MATH] and select Option 2 (\rightarrow Dec).
  • Tracing Function Graphs:
  • Press [GRAPH], then press [TRACE] and enter an $x$-value to inspect point coordinates directly.
  • Absolute Value Input:
  • Press [Y=], press [MATH], arrow right to NUM, and select Option 1 (abs()).

Conceptual Foundation of Functions and Limits

• Function Evaluation vs. Limits:

  • Function evaluation $f(a)$ determines the exact output (or $y$-value) precisely at the single input $x = a$. This represents the destination.
  • Notation $f(3) = 6$ corresponds to the ordered pair $(3, 6)$ on a coordinate graph.
  • A limit evaluates the behavior of function outputs as inputs approach a target value $a$ without ever equaling $a$. This represents the journey.

• Real-World Applications of Limits:

  • Used to analyze system behavior near boundaries where direct evaluation yields undefined results.
  • Black Holes: Physical conditions inside a black hole cannot be directly evaluated, but physical phenomena can be modeled as an observer approaches the event horizon.
  • Population Dynamics: Used to model ecosystem carrying capacity, determining the upper bound value where population growth levels off over time.

Mathematical Definition and Notation of Limits

• Limit Notation: $\lim_{x \rightarrow a} f(x) = L$

  • $x \rightarrow a$ indicates that the independent variable $x$ approaches the real number $a$.
  • $f(x)$ represents the target function under evaluation.
  • $L$ represents the limiting output value ($y$-value) approached by the function.

• Domain Exclusions and Discontinuities:

  • Limits provide descriptions of function behavior at points of discontinuity, such as removable discontinuities (holes), jump discontinuities, and vertical asymptotes (where division by zero occurs).

Types of Limits and One-Sided Limit Behavior

• Left-Hand Limit: $\lim_{x \rightarrow a^-} f(x)$

  • Evaluates the output behavior as $x$ approaches $a$ strictly from values less than $a$ (from the left).
  • Example inputs for $a = 2$: $x = 1.9$, $x = 1.99$, $x = 1.999$

• Right-Hand Limit: $\lim_{x \rightarrow a^+} f(x)$

  • Evaluates the output behavior as $x$ approaches $a$ strictly from values greater than $a$ (from the right).
  • Example inputs for $a = 2$: $x = 2.1$, $x = 2.01$, $x = 2.001$

• Overall (Two-Sided) Limit Existence Criterion:

  • An overall limit $\lim_{x \rightarrow a} f(x)$ exists and equals $L$ if and only if both one-sided limits exist and are equal: $\lim_{x \rightarrow a^-} f(x) = L$$\lim_{x \rightarrow a^+} f(x) = L$$\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = L \implies \lim_{x \rightarrow a} f(x) = L$
  • If $\lim_{x \rightarrow a^-} f(x) \neq \lim_{x \rightarrow a^+} f(x)$, the overall limit does not exist (DNE).

• Multivariable Note (Calculus III Preview):

  • In single-variable calculus, limits approach target points from two directions (left and right).
  • In three-dimensional calculus (Calculus III), limits must approach target points from infinitely many spatial directions and diagonal paths.

Worked Examples: Numerical Evaluation of Limits

• Example 1: Quadratic Function Evaluation

  • Function: $f(x) = 2x^2 - 3x + 1$
  • Evaluate at $x = 2$: $f(2) = 2(2)^2 - 3(2) + 1$$f(2) = 2(4) - 6 + 1$$f(2) = 8 - 6 + 1 = 3$

• Example 2: Rational Function with Removable Discontinuity

  • Function: $f(x) = \frac{x^2 - 1}{x - 1}$
  • Classification: Rational / Fractional Function.
  • Direct Evaluation at $x = 2$: $f(2) = \frac{2^2 - 1}{2 - 1} = \frac{3}{1} = 3$
  • Direct Evaluation at $x = 1$: $f(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0}$
  • Result: $f(1)$ is undefined ($x = 1$ is excluded from the domain).
  • Numerical Limit Construction around $x = 1$:
  • $x = 0.900 \implies f(0.900) = 1.900$
  • $x = 0.990 \implies f(0.990) = 1.990$
  • $x = 0.999 \implies f(0.999) = 1.999$
  • $x = 1.001 \implies f(1.001) = 2.001$
  • $x = 1.010 \implies f(1.010) = 2.010$
  • $x = 1.100 \implies f(1.100) = 2.100$
  • One-Sided Results: $\lim_{x \rightarrow 1^-} f(x) = 2$$\lim_{x \rightarrow 1^+} f(x) = 2$
  • Conclusion: Overall limit exists. $\lim_{x \rightarrow 1} f(x) = 2$
  • Graphical Interpretation: The graph forms a straight line with a single point hole (removable discontinuity) at $(1, 2)$.

• Example 3: Piecewise Function with an Isolated Point

  • Function: $g(x) = \begin{cases} x^2 & \text{if } x \neq 2 \\ 5 & \text{if } x = 2 \end{cases}$
  • Evaluation at non-boundary point $x = 3$: $g(3) = 3^2 = 9$
  • Direct evaluation at boundary point $x = 2$: $g(2) = 5$
  • Limit behavior near $x = 2$ using $g(x) = x^2$ for inputs $x \neq 2$:
  • $x = 1.999 \implies g(1.999) = (1.999)^2 = 3.996001$
  • $x = 2.001 \implies g(2.001) = (2.001)^2 = 4.004001$
  • Limit Conclusion: $\lim_{x \rightarrow 2} g(x) = 4$
  • Comparison: $g(2) = 5$, whereas $\lim_{x \rightarrow 2} g(x) = 4$. The function value and the limit value are completely distinct. The graph features a parabola $y = x^2$ with a hole at $(2, 4)$ and an isolated point defined at $(2, 5)$.

• Example 4: Absolute Value Ratio with Jump Discontinuity

  • Function: $h(x) = \frac{|x|}{x}$
  • Evaluation at $x = 0$: $h(0) = \frac{|0|}{0} = \frac{0}{0} \quad (\text{Undefined})$
  • Left-Hand Numerical Limit ($x \rightarrow 0^-$):
  • $x = -0.001 \implies h(-0.001) = \frac{|-0.001|}{-0.001} = \frac{0.001}{-0.001} = -1$
  • $x = -0.000001 \implies h(-0.000001) = -1$$\lim_{x \rightarrow 0^-} h(x) = -1$
  • Right-Hand Numerical Limit ($x \rightarrow 0^+$):
  • $x = 0.001 \implies h(0.001) = \frac{|0.001|}{0.001} = \frac{0.001}{0.001} = 1$
  • $x = 0.000001 \implies h(0.000001) = 1$$\lim_{x \rightarrow 0^+} h(x) = 1$
  • Conclusion: Because $\lim_{x \rightarrow 0^-} h(x) = -1$ and $\lim_{x \rightarrow 0^+} h(x) = 1$, the overall limit does not exist (DNE). $\lim_{x \rightarrow 0} h(x) = \text{DNE}$
  • Graphical Structure: Piecewise constant horizontal lines with a step jump at $x = 0$

Graphical Approaches to Limits and Types of Discontinuities

• Graphical Analysis Rules:

  • The limit value corresponds strictly to the $y$-level toward which the curve leads from both sides.
  • An isolated point on a graph represents the exact function output $f(a)$, but can never represent the limit value at $a$

• Case 1: Removable Discontinuity with Isolated Point

  • Graph features a continuous path approaching a hole at $(1, 2)$, with a separate filled dot at $(1, 3)$.
  • Function evaluation: $f(1) = 3$
  • Limit evaluation: $\lim_{x \rightarrow 1} f(x) = 2$

• Case 2: Jump Discontinuity

  • Graph approaches $y = 1$ from the left of $x = 0$ and approaches $y = 3$ from the right of $x = 0$
  • One-sided limits: $\lim_{x \rightarrow 0^-} f(x) = 1$$\lim_{x \rightarrow 0^+} f(x) = 3$
  • Overall limit: $\lim_{x \rightarrow 0} f(x) = \text{DNE}$

• Case 3: Vertical Asymptote (Infinite Limit)

  • Graph has a vertical asymptote at $x = 2$.
  • Left branch extends downward indefinitely: $\lim_{x \rightarrow 2^-} f(x) = -\infty$
  • Right branch extends upward indefinitely: $\lim_{x \rightarrow 2^+} f(x) = +\infty$
  • Overall limit: $\lim_{x \rightarrow 2} f(x) = \text{DNE}$

Algebraic Approach: Limits by Direct Substitution

• Direct Substitution Property:

  • If a function $f(x)$ is continuous at $x = a$ (containing no holes, jumps, or asymptotes at that location), the limit as $x$ approaches $a$ equals the function output at $a$: $\lim_{x \rightarrow a} f(x) = f(a)$

• Applicable Function Types:

  • Polynomials: Continuous everywhere; direct substitution works for all real numbers.
  • Rational Functions: Direct substitution works for all $x = a$ where the denominator is non-zero.
  • Radical Functions: Direct substitution works for all values where the radicand is positive (or non-negative for even roots).

• Example: Direct Substitution on a Polynomial

  • Evaluate: $\lim_{x \rightarrow 3} (2x^2 - 5x + 1)$
  • Because $f(x) = 2x^2 - 5x + 1$ is a polynomial, substitute $x = 3$ directly: $\lim_{x \rightarrow 3} (2x^2 - 5x + 1) = f(3)$$f(3) = 2(3)^2 - 5(3) + 1$$f(3) = 2(9) - 15 + 1$$f(3) = 18 - 15 + 1 = 4$$\lim_{x \rightarrow 3} (2x^2 - 5x + 1) = 4$

• Strategy for Algebraic Limits:

  1. Attempt direct substitution first.
  2. If a finite real number is obtained, that value is the limit.
  3. If substitution yields an indeterminate form like $\frac{0}{0}$, algebraic simplification, numerical tables, or graphical analysis must be used.