Complete Study Guide Flashcards: Limits, Counting, Binomial Theorem, Polynomials, Division & Inequalities

Introduction to Limits

• Problem Limits Solve:

  • Imagine driving toward a stop sign and visualizing getting closer and closer ($10$ feet, $5$ feet, $1$ foot, $1$ inch).
  • A limit evaluates what output value ($y$) a function approaches as the input ($x$) gets closer and closer to a specific number.
  • Limits differ from evaluating a function at a point by direct substitution ($f(a)$). A function does not need to be defined at a point (e.g., there can be a hole in the graph) for a limit to exist, because limits only consider behavior near the point, not at the point itself.

• Notation:

  • \n\lim_{x \to a} f(x) = L\n
  • Read as: "The limit of $f(x)$, as $x$ approaches $a$, equals $L$".
  • Meaning: As $x$ approaches $a$ from either side, the $y$ -values of $f(x)$ approach $L$.

• One-Sided Limits:

  • Left-hand limit: \n\lim_{x \to a^-} f(x)\n
  • Evaluates the function value as $x$ approaches $a$ from the left (values slightly less than $a$).
  • Right-hand limit: \n\lim_{x \to a^+} f(x)\n
  • Evaluates the function value as $x$ approaches $a$ from the right (values slightly greater than $a$).

• Two-Sided Limit Existence Rule:

  • Golden Rule: \n\lim_{x \to a} f(x)\n exists if and only if the left-hand limit equals the right-hand limit:
  • \n\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \implies \lim_{x \to a} f(x) = L\n
  • If the left-hand and right-hand limits approach different values (e.g., a jump discontinuity), the two-sided limit Does Not Exist ($\text{DNE}$).

• Reading Limits Off a Graph:

  • Step 1: Locate $x = a$ on the $x$ -axis.
  • Step 2: Trace the curve from the left toward $x = a$ and observe the target $y$ -value (left-hand limit).
  • Step 3: Trace the curve from the right toward $x = a$ and observe the target $y$ -value (right-hand limit).
  • Step 4: Compare values. If equal, that value is the limit. If unequal, write $\text{DNE}$.

• Open vs. Closed Circles:

  • Closed (filled-in) dot: Represents the actual value of the function $f(a)$.
  • Open (hollow) circle: Indicates a gap/hole; the function is not defined at that exact point on the curve.
  • Key Concept: Limits ignore where the function is actually plotted at $x = a$ (or if it is undefined). Limits only measure the height the curve approaches from both sides.

• Limit vs. Actual Function Value:

  • \n\lim_{x \to a} f(x)\n: The height the curve approaches near $x = a$ .
  • $f(a)$: The actual $y$ -value plotted at $x = a$ .
  • Possible scenarios:
  • Same value: Continuous curve.
  • Different values: A point is redefined away from the curve.
  • Limit exists while $f(a)$ is undefined: An open circle exists with no solid point at $x = a$ .

• Explaining Why a Limit Does Not Exist:

  • Example 1: "The left-hand limit is $2$ but the right-hand limit is $5$, so they disagree — $\text{DNE}$".
  • Example 2: "As $x$ approaches this value, the curve shoots up toward infinity without leveling off at any single height — $\text{DNE}$".

• Worked Example for Graphical Limits:

  • Given a curve approaching height $3$ from the left with an open circle at $(-2, 3)$, continuing past $x = -2$ from height $3$ on the right, and a separate solid dot at $(-2, 1)$.
  • Left-hand limit: \n\lim_{x \to -2^-} f(x) = 3\n
  • Right-hand limit: \n\lim_{x \to -2^+} f(x) = 3\n
  • Two-sided limit: \n\lim_{x \to -2} f(x) = 3\n
  • Function value: $f(-2) = 1$

• Quick Checklist for Finding Limits from Graphs:

  • 1. Check if the prompt asks for $f(a)$ (solid dot) or \n\lim_{x \to a} f(x)\n (tracing the curve).
  • 2. Identify if the limit is one-sided ($-$ or $+$) or two-sided.
  • 3. Trace from the appropriate side(s) ignoring open/closed dots.
  • 4. For two-sided limits, confirm if left and right values match.
  • 5. Re-read to confirm whether a function value or limit was requested.

Limits at Infinity & Infinite Limits (Asymptotes)

• Distinguishing Two Limit Types:

  • Limit at Infinity: \n\lim_{x \to \infty} f(x)\n or \n\lim_{x \to -\infty} f(x)\n. Asks what height $y$ levels off to as $x$ moves far right or far left. Describes end behavior and horizontal asymptotes.
  • Infinite Limit: \n\lim_{x \to c} f(x) = \infty\n or $-\infty$. Asks what happens near a finite value $x = c$ when the curve shoots vertically without bound. Describes vertical asymptotes.

• Horizontal Asymptotes:

  • If the curve flattens toward height $L$ as $x \to \infty$ or $x \to -\infty$, then $y = L$ is a horizontal asymptote.
  • Worked Example: A rational function flattens hugging $y = 0$ far to the left and far to the right.
  • \n\lim_{x \to -\infty} f(x) = 0\n
  • \n\lim_{x \to \infty} f(x) = 0\n
  • Horizontal asymptote is at $y = 0$ .

• Vertical Asymptotes:

  • If the curve goes to $\infty$ or $-\infty$ near $x = c$, $x = c$ is a vertical asymptote.
  • Worked Example: Dashed line at $x = 2$ . Left side plunges downward; right side shoots upward.
  • \n\lim_{x \to 2^-} f(x) = -\infty\n
  • \n\lim_{x \to 2^+} f(x) = \infty\n
  • Two-sided limit: \n\lim_{x \to 2} f(x) = \text{DNE}\n
  • Both sides shooting in the same direction:
  • If \n\lim_{x \to c^-} f(x) = \infty\n and \n\lim_{x \to c^+} f(x) = \infty\n, then \n\lim_{x \to c} f(x) = \infty\n .

• Step-by-Step Method for Graph Reading:

  • 1. Identify limit type (ordinary finite, infinite limit, or limit at infinity).
  • 2. Focus on relevant region (far edges for $x \to \pm\infty$, near $x = c$ for finite limits).
  • 3. Describe trend (finite height $L$, unbounded $\pm\infty$, or $\text{DNE}$).
  • 4. Write answer using correct mathematical notation.

• Sketching a Graph from Limit Properties:

  • Worked Example Clues:
  • \n\lim_{x \to \infty} f(x) = -1\n
  • \n\lim_{x \to 3^+} f(x) = \infty\n
  • $f(-2) = 0$
  • \n\lim_{x \to -5^-} f(x) = -\infty\n
  • \n\lim_{x \to 3} f(x) = \infty\n
  • \n\lim_{x \to -5} f(x) = \text{DNE}\n
  • Translation of Clues:
  • Far right flattens along horizontal line $y = -1$ .
  • Right side of $x = 3$ shoots to $\infty$ .
  • Two-sided limit at $x = 3$ is $\infty$, so left side of $x = 3$ also shoots to $\infty$ .
  • Left side of $x = -5$ dives to $-\infty$ .
  • Two-sided limit at $x = -5$ is $\text{DNE}$, so right side of $x = -5$ must shoot to $\infty$ .
  • Plot a solid point at $(-2, 0)$, which lies in the middle region between $x = -5$ and $x = 3$ .

Counting Principles, Permutations, Combinations & the Binomial Theorem

• Fundamental Counting Principle (FCP):

  • Rule: If event 1 occurs in $m$ ways and event 2 occurs in $n$ ways, both occurring in sequence can happen in $m \times n$ ways.
  • Multi-step general rule: \n n_1 \times n_2 \times n_3 \times \dots\n
  • Method: Draw an empty box for each decision, write the number of available options in each box, and multiply.
  • Meal Example: $4$ main courses, $3$ drinks, $2$ desserts.
  • \n4 \times 3 \times 2 = 24\text{ possible meals}\n
  • Ice Cream Example: $3$ cone types, $31$ flavors.
  • \n3 \times 31 = 93\text{ possible cones}\n
  • License Plate Example (3 letters, 3 digits - repetition allowed):
  • \n26 \times 26 \times 26 \times 10 \times 10 \times 10 = 17,576,000\n
  • License Plate Example (letters cannot repeat, digits can):
  • \n26 \times 25 \times 24 \times 10 \times 10 \times 10 = 15,600,000\n
  • Race Example (6 runners, ranking 1st through 6th):
  • \n6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720\n

• Factorials:

  • Definition: \n n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\n
  • Example: \n5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\n
  • Convention: \n0! = 1\n

• Permutations (Order Matters):

  • Definition: An arrangement of objects where the specific sequence or assigned role alters the outcome (e.g., assigning President, Vice President, Secretary).
  • Permutations of all $n$ distinct objects: \n n!\n
  • Permutations of $n$ objects choosing/arranging only $r$:
  • Formula: \n P(n, r) = \frac{n!}{(n - r)!}\n
  • Club Example: Choose President, VP, Secretary from $9$ members.
  • \n P(9, 3) = \frac{9!}{(9 - 3)!} = \frac{9!}{6!} = 9 \times 8 \times 7 = 504\n
  • Raffle Example: $20$ tickets, $4$ distinct prizes (car, motorcycle, bicycle, skateboard).
  • \n P(20, 4) = \frac{20!}{(20 - 4)!} = \frac{20!}{16!} = 20 \times 19 \times 18 \times 17 = 116,280\n

• Distinguishable Permutations (Identical Copies):

  • Formula when arranging $n$ total objects where $n_1, n_2, \dots, n_k$ are identical copies:
  • \n\text{Distinguishable Permutations} = \frac{n!}{n_1! n_2! n_3! \dots n_k!}\n
  • Worked Example: Arrange $15$ balls in a row ($4$ red, $3$ yellow, $6$ black, $2$ blue).
  • \n\frac{15!}{4! 3! 6! 2!} = 6,306,300\n

• Seating Restrictions ("Together / Apart" Trick):

  • Two people standing together: Glue them into $1$ combined block.
  • $12$ people total, Jane and John together: Arrange $11$ items ($11!$), then multiply by $2$ for Jane-John vs. John-Jane ordering inside the block.
  • \n2 \times 11! = 79,833,600\n
  • Two people standing apart:
  • Subtract grouped arrangements from total arrangements:
  • \n12! - (2 \times 11!) = 399,168,000\n

• Combinations (Order Does NOT Matter):

  • Definition: Selection of $r$ items from $n$ options where sequence or placement does not change the outcome.
  • Formula: \n C(n, r) = \frac{n!}{r!(n - r)!}\n
  • Derivation: Takes permutation count $P(n, r)$ and divides by $r!$ to eliminate duplicate orderings.
  • Unranked Committee Example: Choose $3$ members from $9$ .
  • \n C(9, 3) = \frac{9!}{3! 6!} = 84\n
  • Raffle Example: $4$ identical prizes chosen from $20$ tickets.
  • \n C(20, 4) = \frac{20!}{4! 16!} = 4,845\n

• Subsets of a Set:

  • A set with $n$ elements has $2^n$ total subsets (including empty set and full set).
  • Pizza Topping Example: $16$ optional toppings.
  • \n2^{16} = 65,536\text{ possible topping combinations}\n

• Permutation vs. Combination Decision Matrix:

  • Swapping order changes outcome: Permutation $P(n, r)$
  • Swapping order gives identical outcome: Combination $C(n, r)$

• Multi-Part / Multi-Step Problems:

  • Scouting party: Choose $3$ women from $15$, and $2$ men from $10$ .
  • \n C(15, 3) \times C(10, 2) = 455 \times 45 = 20,475\n
  • Mixed Committee: $20$ students total; choose $3$ assigned roles and $4$ unranked general members.
  • Stage 1 (3 roles from 20): $P(20, 3) = 6,840$
  • Stage 2 (4 general members from remaining 17): $C(17, 4) = 2,380$
  • Total: \n6,840 \times 2,380 = 16,279,200\n

• The Binomial Theorem:

  • Pascal's Triangle Row Coefficients:
  • $n = 0$: $1$
  • $n = 1$: $1 \quad 1$
  • $n = 2$: $1 \quad 2 \quad 1$
  • $n = 3$: $1 \quad 3 \quad 3 \quad 1$
  • $n = 4$: $1 \quad 4 \quad 6 \quad 4 \quad 1$
  • $n = 5$: $1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1$
  • $n = 6$: $1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1$
  • $n = 7$: $1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1$
  • Properties of Expansion $(a + b)^n$:
  • 1. Contains $n + 1$ total terms.
  • 2. First term is $a^n$, last term is $b^n$ .
  • 3. Exponent on $a$ decreases by $1$ each term; exponent on $b$ increases by $1$ .
  • 4. Sum of exponents in any single term equals $n$ .
  • Expansion Example using Pascal's Triangle: Expand $(2 - 3x)^5$
  • Set $a = 2$, $b = -3x$. Use row 5 coefficients ($1, 5, 10, 10, 5, 1$ ):
  • \n(2 - 3x)^5 = 1(2)^5 + 5(2)^4(-3x) + 10(2)^3(-3x)^2 + 10(2)^2(-3x)^3 + 5(2)^1(-3x)^4 + 1(-3x)^5\n
  • \n(2 - 3x)^5 = 32 - 240x + 720x^2 - 1080x^3 + 810x^4 - 243x^5\n

• Binomial Coefficients:

  • Formula: \n\binom{n}{r} = \frac{n!}{r!(n - r)!}\n
  • Examples:
  • \n\binom{9}{4} = \frac{9!}{4! 5!} = 126\n
  • \n\binom{100}{3} = \frac{100!}{3! 97!} = 161,700\n
  • Symmetry Identity: \n\binom{n}{r} = \binom{n}{n - r}\n
  • Addition Rule (Pascal's Rule): \n\binom{k}{r} + \binom{k}{r + 1} = \binom{k + 1}{r + 1}\n

• Full Binomial Expansion Formula:

  • \n(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n\n
  • Compact Summation Form: \n(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\n
  • Worked Example: Expand $(x + y)^4$
  • Coefficients: $\binom{4}{0}=1, \binom{4}{1}=4, \binom{4}{2}=6, \binom{4}{3}=4, \binom{4}{4}=1$
  • \n(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\n
  • Worked Example: Expand $(\sqrt{x} - 1)^8$
  • Row 8 coefficients: $1, 8, 28, 56, 70, 56, 28, 8, 1$
  • Substitute $a = x^{1/2}$ and $b = -1$ :
  • \n(\sqrt{x} - 1)^8 = x^4 - 8x^{7/2} + 28x^3 - 56x^{5/2} + 70x^2 - 56x^{3/2} + 28x - 8x^{1/2} + 1\n

• Finding One Specific Term:

  • The term containing $a^r$ in $(a + b)^n$ is given by:
  • \n\text{Term} = \binom{n}{n - r} a^r b^{n - r}\n
  • Example: Find term containing $x^5$ in $(2x + y)^{20}$
  • $a = 2x$, $b = y$, $n = 20$, $r = 5$
  • \n\binom{20}{15}(2x)^5 y^{15} = 15,504 \times 32x^5 y^{15} = 496,128 x^5 y^{15}\n
  • Example: Find coefficient of $x^8$ in $\left(x^2 + \frac{1}{x}\right)^{10}$
  • General term: \n\binom{10}{10 - r} (x^2)^r \left(\frac{1}{x}\right)^{10 - r} = \binom{10}{10 - r} x^{3r - 10}\n
  • Set exponent to $8$: $3r - 10 = 8 \implies 3r = 18 \implies r = 6$
  • Coefficient: \n\binom{10}{4} = 210\n

• Induction Proof Logic:

  • Base Case: Confirm formula for $n = 1$: $(a + b)^1 = a + b$ .
  • Inductive Step: Assume true for $n = k$ . Multiply by $(a + b)$ to form $(a + b)^{k+1}$. Combining like terms yields Pascal's addition identity $\binom{k}{r} + \binom{k}{r+1} = \binom{k+1}{r+1}$, completing the proof.

Polynomial Functions, End Behavior & Graphing

• Definition of Polynomial Function:

  • \n P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\n
  • Requirements: Coefficients $a_i$ are fixed numbers; all exponents $n$ must be non-negative integers ($0, 1, 2, 3, \dots$).

• Degree and Leading Term:

  • Degree of term: Exponent on the variable.
  • Degree of polynomial ($n$): Highest exponent in polynomial.
  • Leading term: Term with highest degree ($a_n x^n$).
  • Leading coefficient: $a_n$ .

• End Behavior Summary Table:

  • Even Degree, $a_n > 0$: Both ends go up ($\lim_{x \to \infty} P(x) = \infty$, $\lim_{x \to -\infty} P(x) = \infty$).
  • Even Degree, $a_n < 0$: Both ends go down ($\lim_{x \to \infty} P(x) = -\infty$, $\lim_{x \to -\infty} P(x) = -\infty$).
  • Odd Degree, $a_n > 0$: Right goes up, left goes down ($\lim_{x \to \infty} P(x) = \infty$, $\lim_{x \to -\infty} P(x) = -\infty$).
  • Odd Degree, $a_n < 0$: Right goes down, left goes up ($\lim_{x \to \infty} P(x) = -\infty$, $\lim_{x \to -\infty} P(x) = \infty$).

• Derivation of Leading Term Dominance:

  • Factor $a_n x^n$ out of $P(x) = 3x^3 + 5x^2 - 7$ :
  • \n P(x) = 3x^3 \left(1 + \frac{5}{3x} - \frac{7}{3x^3}\right)\n
  • As $|x| \to \infty$, fractions $\frac{5}{3x}$ and $\frac{7}{3x^3}$ go to $0$ .
  • Therefore, $P(x) \to 3x^3 \cdot 1 = 3x^3$ .

• Roots (Zeros) and Multiplicity:

  • Root: Any $a$ where $P(a) = 0$ .
  • Multiplicity: The exponent $m$ on factor $(x - a)^m$ .
  • Odd Multiplicity ($1, 3, 5, \dots$): Graph crosses the $x$ -axis at that root.
  • Even Multiplicity ($2, 4, 6, \dots$): Graph touches the $x$ -axis and bounces off (tangent to axis).

• Root Count & Existence Theorems:

  • A polynomial of degree $n$ has exactly $n$ complex roots (counting multiplicities).
  • Odd-degree polynomials are guaranteed to have at least one real root (by continuity and opposite end behaviors).

• Relative Extrema (Turning Points):

  • A polynomial of degree $n$ has at most $n - 1$ relative extrema (turning points).

• Polynomial Sketching Checklist:

  • 1. Identify degree and leading coefficient to determine end behavior.
  • 2. Factor completely to determine all roots and multiplicities.
  • 3. Classify each root as cross-through (odd) or bounce-off (even).
  • 4. Confirm turning points do not exceed $n - 1$ .
  • 5. Connect endpoints and roots continuously from left to right.

Polynomial Division

• Division Algorithm:

  • Fraction form: \n\frac{P(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}\n
  • Product form: \n P(x) = q(x) \cdot d(x) + r(x)\n
  • Degree restriction: \n0 \le \text{deg}(r) < \text{deg}(d)\n

• Polynomial Long Division Worked Example:

  • Divide $4x^4 + 7x^2 - 34x + 15$ by $x^2 + 2x + 5$ .
  • Step 0: Add zero placeholder: $4x^4 + 0x^3 + 7x^2 - 34x + 15$ .
  • Step 1: Divide leading terms $4x^4 \div x^2 = 4x^2$ .
  • Step 2: Multiply $4x^2(x^2 + 2x + 5) = 4x^4 + 8x^3 + 20x^2$ .
  • Step 3: Subtract: $(4x^4 + 0x^3 + 7x^2) - (4x^4 + 8x^3 + 20x^2) = -8x^3 - 13x^2$ . Bring down $-34x$ .
  • Step 4: Divide $-8x^3 \div x^2 = -8x$ . Multiply $-8x(x^2 + 2x + 5) = -8x^3 - 16x^2 - 40x$ . Subtract: $(-8x^3 - 13x^2 - 34x) - (-8x^3 - 16x^2 - 40x) = 3x^2 + 6x$ . Bring down $15$ .
  • Step 5: Divide $3x^2 \div x^2 = 3$ . Multiply $3(x^2 + 2x + 5) = 3x^2 + 6x + 15$ . Subtract: $(3x^2 + 6x + 15) - (3x^2 + 6x + 15) = 0$ .
  • Final Quotient: $4x^2 - 8x + 3$, Remainder: $0$ .
  • Result: \n\frac{4x^4 + 7x^2 - 34x + 15}{x^2 + 2x + 5} = 4x^2 - 8x + 3\n

• Synthetic Division Rules:

  • Applies strictly to linear divisors of the form $x - c$ .
  • Setup: Write coefficients of dividend (with zero placeholders) and $c$ on the left.
  • Worked Example: Divide $x^3 + 2x^2 - 10x + 1$ by $x - 2$ ($c = 2$ ):
  • Table setup:
    • $c = 2$
    • Row 1: $1, 2, -10, 1$
    • Row 2: $\quad 2, 8, -4$
    • Row 3: $1, 4, -2, -3$
  • Row 3 interpretation: Quotient coefficients $1, 4, -2$ (degree $2$) and remainder $-3$ .
  • Fraction Form: \n\frac{x^3 + 2x^2 - 10x + 1}{x - 2} = x^2 + 4x - 2 + \frac{-3}{x - 2}\n
  • Product Form: \n x^3 + 2x^2 - 10x + 1 = (x^2 + 4x - 2)(x - 2) - 3\n

• Additional Synthetic Division Examples:

  • Example A: Divide $x^3 + 7x^2 - x + 11$ by $x - 5$ ($c = 5$ ):
  • Synthetic entries: $5 \mid 1 \quad 7 \quad -1 \quad 11$
  • Operations: $1 \to 1\cdot 5=5 \to 7+5=12 \to 12\cdot 5=60 \to -1+60=59 \to 59\cdot 5=295 \to 11+295=306$
  • Fraction form: \n\frac{x^3 + 7x^2 - x + 11}{x - 5} = x^2 + 12x + 59 + \frac{306}{x - 5}\n
  • Product form: \n x^3 + 7x^2 - x + 11 = (x^2 + 12x + 59)(x - 5) + 306\n
  • Example B: Divide $4x^3 - 2x + 3$ by $x + 1$ ($c = -1$, include $0x^2$ ):
  • Synthetic entries: $-1 \mid 4 \quad 0 \quad -2 \quad 3$
  • Operations: $4 \to 4(-1)=-4 \to 0-4=-4 \to -4(-1)=4 \to -2+4=2 \to 2(-1)=-2 \to 3-2=1$
  • Product form: \n4x^3 - 2x + 3 = (4x^2 - 4x + 2)(x + 1) + 1\n

Solving Polynomial Inequalities

• General Method:

  • Step 1: Rearrange inequality so one side is compared to $0$ .
  • Step 2: Factor completely to identify all real roots and multiplicities.
  • Step 3: Plot roots on a number line to form test intervals.
  • Step 4: Determine polynomial sign in each interval.
  • Step 5: Select matching intervals. Exclude roots for strict inequalities ($>, <$); include roots for non-strict inequalities ($\ge, \le$).

• Interval Sign Determination Methods:

  • Method A (Test-Point Method): Choose a test value inside each interval, plug into factored expression, and evaluate net sign.
  • Method B (End-Behavior + Multiplicity Shortcut):
  • Determine sign of rightmost interval using leading term end behavior.
  • Move right-to-left across roots:
    • Odd multiplicity root: Sign flips.
    • Even multiplicity root: Sign stays the same.

• Worked Examples:

  • Example 1: Solve $x^2 - x - 6 > 0$
  • Factored: $(x - 3)(x + 2) > 0$
  • Roots: $x = -2$ (mult 1), $x = 3$ (mult 1)
  • End behavior: Positive as $x \to \infty$
  • Signs across intervals: $(3, \infty) \to +$, $(-2, 3) \to -$, $(-\infty, -2) \to +$
  • Solution: $x < -2 \text{ or } x > 3$

• Example 2: Solve $(x - 1)^2(x + 3) \le 0$
  • Roots: $x = 1$ (mult 2), $x = -3$ (mult 1)
  • End behavior: Positive as $x \to \infty$
  • Signs across intervals: $(1, \infty) \to +$, $(-3, 1) \to +$, $(-\infty, -3) \to -$
  • Solution: $x \le -3 \text{ or } x = 1$

• Example 3: Solve $(x + 2)(x - 1)(x - 4) < 0$
  • Roots: $x = -2, 1, 4$ (all mult 1)
  • End behavior: Positive as $x \to \infty$
  • Signs across intervals: $(4, \infty) \to +$, $(1, 4) \to -$, $(-2, 1) \to +$, $(-\infty, -2) \to -$
  • Solution: $x < -2 \text{ or } 1 < x < 4$

Quick-Reference Formula Sheet & Common Mistakes

• Formula Sheet: Limits:

  • Two-sided limit existence: \n\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\n
  • Limit at infinity: \n\lim_{x \to \pm\infty} f(x) = L \implies y = L\text{ (Horizontal Asymptote)}\n
  • Infinite limit: \n\lim_{x \to c} f(x) = \pm\infty \implies x = c\text{ (Vertical Asymptote)}\n

• Formula Sheet: Counting & Algebra:

  • Fundamental Counting Principle: \n n_1 \times n_2 \times n_3 \times \dots\n
  • Factorials: \n n! = n(n-1)(n-2)\dots 1, \quad 0! = 1\n
  • Permutations: \n P(n, r) = \frac{n!}{(n - r)!}\n
  • Distinguishable Permutations: \n\frac{n!}{n_1! n_2! \dots n_k!}\n
  • Combinations: \n C(n, r) = \frac{n!}{r!(n - r)!}\n
  • Total Subsets: \n2^n\n
  • Binomial Coefficient: \n\binom{n}{r} = \frac{n!}{r!(n - r)!}\n
  • Binomial Theorem: \n(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\n
  • General Binomial Term: \n\binom{n}{n - r} a^r b^{n - r}\n

• Formula Sheet: Polynomials & Division:

  • End behavior: Dominant leading term $a_n x^n$
  • Roots: Multiplicity odd $\implies$ cross; Multiplicity even $\implies$ bounce
  • Max relative extrema: $n - 1$
  • Division algorithm: \n P(x) = q(x)d(x) + r(x)\n
  • Synthetic division condition: Divisor must be $x - c$

• Common Mistakes to Avoid:

  • Limits: Confusing \n\lim_{x \to a} f(x)\n with $f(a)$. Ignoring open vs. closed dots. Forgetting to verify both sides for two-sided limits.
  • Limits at Infinity: Confusing horizontal asymptote limits ($x \to \infty$) with vertical asymptote limits ($f(x) \to \infty$).
  • Counting: Using permutations when order does not matter. Setting $0! = 0$ instead of $1$. Forgetting to apply FCP to multiply multi-stage combination results.
  • Binomial Theorem: Forgetting sign alternation when $b$ is negative. Reversing exponents on $a$ and $b$ in term formulas.
  • Polynomial Graphs & Division: Forgetting zero placeholders for missing terms. Using synthetic division for non-linear divisors.
  • Inequalities: Failing to set one side to $0$ before factoring. Forgetting that even multiplicity roots do not change signs. Omitting isolated solution points in non-strict inequalities.