Page 2: Advanced Limit Calculations and Proofs

This page delves into advanced limit calculations, particularly focusing on fundamental limits and their proofs, including the famous limit of $1 + 1/n$ⁿ approaching e.

Definition: The limiti fondamentali include crucial limits such as lim(x→0) sin(x)/x = 1 and lim(n→∞) $1 + 1/n$ⁿ = e.

Example: A key demonstration shows that sin(x)/x < 1 for x approaching 0, using geometric arguments involving sector areas.

Highlight: The proof of lim(x→0) sin(x)/x = 1 uses a geometric approach comparing areas of triangles, sectors, and regions bounded by trigonometric functions.

Quote: "sinx < x < tanx" represents a fundamental inequality used in proving trigonometric limits.

Page 1: Fundamental Concepts and Theorems

This page introduces essential trigonometric identities, logarithmic properties, and the fundamental concepts of limits, including the comparison theorem and indeterminate forms.

Definition: The teorema del confronto states that if g(x) ≤ f(x) ≤ h(x) for all x in an interval, and the limits of g(x) and h(x) both equal l as x approaches x₀, then the limit of f(x) also equals l.

Vocabulary: Forme indeterminate refers to limit expressions that cannot be directly evaluated, such as 0/0, ∞-∞, and 0·∞.

Example: For trigonometric functions, key identities include sin(2x) = 2sinxcosx and cos(2x) = cos²x - sin²x.

Highlight: The page presents four distinct cases for limit evaluation:

1. When x₀ and l are finite
2. When x₀ is finite and l is infinite
3. When x₀ is infinite and l is finite
4. When both x₀ and l are infinite