Page 6: Advanced Limit Techniques

This page covers advanced techniques for solving complex limit problems, particularly those involving indeterminate forms.

Key techniques discussed include:

1. Finding equivalent fractions by using the least common multiple (LCM)
2. Multiplying numerator and denominator by conjugates
3. Analyzing the behavior of highest degree terms as x approaches infinity

Example: For the limit of (x² + 2x + 3x³ - 6x²) / (3x² - 1)(x - 3) as x approaches infinity, the focus is on the highest degree terms in both numerator and denominator.

The page also covers limits involving square roots and the product of functions.

Highlight: Special attention is given to transforming indeterminate forms like ∞ - ∞ into more manageable forms like 0/0.

The problem-solving approach emphasizes identifying the dominant terms in expressions as x approaches infinity and using algebraic manipulations to simplify complex expressions.

Page 3: Bolzano's Theorem and Function Intersections

This page focuses on applications of Bolzano's Theorem to prove the existence of roots and intersection points of functions.

Definition: Bolzano's Theorem states that if a continuous function f(x) has opposite signs at the endpoints of a closed interval [a,b], then there exists at least one point c in (a,b) where f(c) = 0.

The page presents two main problems:

1. Proving that the function g(x) = x + 1/x has at least one point where it equals 2.
2. Demonstrating that the functions g(x) = x + 1 and h(x) = x² + 3x intersect.

Example: To prove the intersection of g(x) and h(x), a new function f(x) = g(x) - h(x) is constructed, and Bolzano's Theorem is applied to show that f(x) has a root.

The page also includes a method for approximating the x-coordinate of the intersection point to within one decimal place accuracy.

Highlight: The problem-solving approach emphasizes constructing appropriate functions and applying theoretical results to practical situations.

Page 4: Continuity and Absolute Value Functions

This page explores continuity of functions involving [absolute values](https://knowunity.es/knows/matemticas-valor-absoluto-y-opuesto-de-nmeros-enteros-080f8b2a-ca0c-4c2c-80f2-e114a2deacae?utm_content=seo_link) and piecewise definitions.

Definition: The absolute value of a function is defined as |f(x)| = f(x) if f(x) ≥ 0, and -f(x) if f(x) < 0.

Key points covered:

1. Checking continuity at boundary points of piecewise functions
2. Analyzing continuity of functions involving absolute values
3. Identifying types of discontinuities (e.g., jump discontinuity)

Example: For the function f(x) = |3-x|, it's broken down into two cases: 3-x if x ≤ 3, and -(3-x) if x > 3.

The page emphasizes the importance of carefully analyzing functions at points where their definition changes, such as at x = 0 for absolute value functions.

Highlight: Polynomial functions are noted to be continuous over their entire domain, which is an important property in continuity analysis.