Limit Notation and Meaning

The abbreviation 'lim' stands for 'limit.' It signals that we are asking: what value does the expression to the right approach under the condition written below the symbol? Writing lim alone is incomplete; it always needs a subscript and an expression.

Below 'lim' we write the approach condition, most commonly x → c, read as 'x approaches c.' This means x takes values closer and closer to c from both sides — but x never actually equals c. The arrow (→) emphasizes the dynamic process of approaching, not arriving.

To the right of 'lim' we write the function or expression whose behavior we are studying. For example, lim(x→2) (x² − 4)/(x − 2). Notice that substituting x = 2 directly gives 0/0, which is undefined. The limit asks what value the expression approaches as x gets close to 2, not at x = 2.

When the limit exists, we set it equal to a finite value L. Writing lim(x→c) f(x) = L means: for every desired degree of closeness to L (no matter how small), we can always find x sufficiently close (but not equal) to c so that f(x) is that close to L. This informal idea is made precise by the formal epsilon-delta definition.

Sometimes f(x) approaches different values from the left and the right. We use lim(x→c⁻) f(x) = L₁ for the left-hand limit (x approaches c from values less than c) and lim(x→c⁺) f(x) = L₂ for the right-hand limit. The two-sided limit lim(x→c) f(x) exists if and only if both one-sided limits exist and are equal: L₁ = L₂ = L.

The notation lim(x→∞) f(x) = L means f(x) approaches L as x grows without bound. Similarly, lim(x→−∞) f(x) = L describes behavior as x decreases without bound. These limits describe horizontal asymptotes. For example, lim(x→∞) 1/x = 0 because 1/x shrinks toward zero as x becomes very large.

A limit fails to exist (written DNE) in three main situations: (1) the left-hand and right-hand limits are unequal; (2) f(x) grows without bound near c (we may write lim = ∞ as a description, not a finite value); (3) f(x) oscillates endlessly and never settles, as in lim(x→0) sin(1/x). Recognizing non-existence is just as important as finding limits that do exist.

• \lim — Limit operator (dimensionless): Indicates we are taking a limit; always paired with a subscript describing the approach condition.
• x \to c — Approach condition (dimensionless): The variable x approaches the value c; x gets arbitrarily close to c but never equals c.
• f(x) — Function expression (varies): The function whose output we are tracking as x approaches c.
• L — Limit value (varies): The unique finite value (if it exists) that f(x) approaches as x → c.

Problem. Find lim(x→3) (x² − 9)/(x − 3) and explain why direct substitution fails but the limit still exists.

Answer. The limit equals 6.

• Continuity of Functions — Continuity is defined using limits: a function is continuous at c if lim(x→c) f(x) = f(c). Understanding limit notation is prerequisite to understanding continuity.
• Limit Laws and Evaluation Techniques — Once the notation and meaning of limits are clear, the next step is learning algebraic rules (sum, product, quotient laws) and techniques such as factoring, rationalization, and L'Hôpital's rule to evaluate limits efficiently.
• Definition of the Derivative — The derivative f'(x) is formally defined as lim(h→0) [f(x+h)−f(x)]/h. Limit notation is the language in which the derivative is expressed and understood.
• Infinite Limits and Asymptotes — Vertical and horizontal asymptotes are described precisely using limit notation, extending the concept of limits to situations where function values or inputs grow without bound.