Limit Notation

Direct Answer

The limit notation lim_{x → a} f(x) = L means that as the variable x gets arbitrarily close to the point a, the value of the function f(x) approaches the number L.

Informal Interpretation

Imagine sliding along the graph of f(x) toward the vertical line x = a. Even if f(a) is undefined or different, the values of f(x) can still settle arbitrarily close to L. The limit captures this “getting closer” behavior without requiring actual equality at x = a.

Formal (ε–δ) Definition

1. For every ε > 0,
2. There exists a δ > 0,
3. Such that whenever 0 < |x − a| < δ,
4. It follows that |f(x) − L| < ε.

Variants of Limit Notation

• One-sided limits
  • lim_{x → a⁻} f(x) = L (approach from left)
  • lim_{x → a⁺} f(x) = L (approach from right)
• Limits at infinity
  • lim_{x → ∞} f(x) = L
  • lim_{x → −∞} f(x) = L
• Infinite limits
  • lim_{x → a} f(x) = ∞ means f(x) grows arbitrarily large as x approaches a.
• Sequence limits
  • lim_{n → ∞} aₙ = L applies the same idea to a sequence (aₙ).

Examples

1. lim_{x → 2} (3x + 1) = 7
2. lim_{x → 0} sin(x)/x = 1
3. lim_{x → ∞} 1/x = 0
4. lim_{x → 1⁻} ln(x) = 0

Common Notational Conventions

• Omit writing lim when context is clear in proofs (e.g., “As x → 0, sin x ∼ x”).
• Use x → a⁺ or a⁻ for boundary behavior on an interval.
• In multiple variables: lim_{(x,y) → (a,b)} f(x,y) = L.

Next Steps

1. Techniques for evaluating limits (algebraic simplification, squeeze theorem, L’Hôpital’s rule).
2. How limits define continuity and the derivative.
3. Extensions to metric spaces and topology, where lim generalizes via neighborhoods.
4. Series and infinite products: ∑_n=0^∞ aₙ converges if lim_{N → ∞} ∑_n=0^N aₙ exists.