n → ∞
The Horizon
Limits, Continuity & The Infinite Approach
The Human Problem
We have studied functions by evaluating them at exact points. But what if a point is unreachable? What if a formula breaks down at x = a? We must shift our perspective. Instead of asking what a function is at a point, we ask: What value does it approach as we get arbitrarily close? This is the conceptual leap into Calculus.
APPROACH →
LIMIT →
ONE-SIDED LIMITS →
LIMIT LAWS →
INDETERMINATE FORMS →
INFINITE LIMITS →
ASYMPTOTES →
CONTINUITY →
THE DERIVATIVE
The 58 Horizon Cells
From Intuitive Approach to Epsilon-Delta Rigor
CELL 01-06
Foundations of Approach
x → a⁻
x → a⁺
- The Idea of Approaching
- Intuitive Limit Definition
- Value vs Limit Distinction
- The Deleted Point
- Numerical & Graphical Limits
CELL 07-10
One-Sided Limits
L
R
- Left-Hand Limit (x → a⁻)
- Right-Hand Limit (x → a⁺)
- Two-Sided Limit Existence
- Non-Existence (Jumps)
CELL 11-19
Limit Laws
lim[f(x) + g(x)]
lim f(x) + lim g(x)
- Direct Substitution
- Sum, Diff, Product, Quotient
- Limits of Polynomials
- Rational Limits & Denominators
CELL 20-24
Indeterminate Forms
- The Meaning of 0/0
- Factorization Method
- Cancelling Common Factors
- Rationalization (Conjugates)
- Limits Involving Radicals
CELL 25-28
Standard Limits
lim
sin xx
= 1
- The Fundamental Trig Limit
- Related Trig Limits
- Exponential Limit Preview
- Logarithmic Limit Preview
CELL 29-30
The Squeeze Theorem
- g(x) ≤ f(x) ≤ h(x)
- Trapping the Function
- Geometric Limit Applications
CELL 31-38
Infinite Behavior
- Limits at Infinity (x → ±∞)
- Infinite Limits (f(x) → ±∞)
- Horizontal & Vertical Asymptotes
- Polynomial/Rational Dominance
CELL 39-49
Continuity
- Formal Continuity Definition
- Removable & Jump Discontinuities
- Continuous Functions Library
- Piecewise & Interval Continuity
CELL 50-55
Theorems & Rigor
- Intermediate Value Theorem
- Root Existence & Approximation
- Local vs Global Behavior
- Epsilon-Delta (ε-δ) Definition
CELL 56-58
The Frontier
- Limits as Calculus Foundation
- Secant Rotating to Tangent
- Preparing for the Derivative
Exit Portal
We have learned how to study a function's destination, even if it never arrives. Now, the question changes. Instead of asking "Where is the function heading?" we ask: "How fast is it changing at this exact instant?"