The Limit of a Function: Intuition, One-Sided Limits, and Notation

The derivative of an option price with respect to spot is the option's delta — the hedge ratio. That derivative is defined as a limit: the ratio of a price change to a spot move, as the spot move shrinks to zero. Before any derivative can be computed, the concept of a limit must be in place. This note builds that concept from two physical problems — the tangent problem and the velocity problem — then introduces the informal definition, one-sided limits, and the ways a limit can fail to exist.

• motivate the limit concept from the tangent-line and instantaneous-velocity problems;
• state the informal definition of $\lim_{x \to a} f(x) = L$ and use it to guess limits from tables and graphs;
• define left-hand and right-hand limits and state the condition $\lim_{x\to a} f = L \Leftrightarrow \lim_{x\to a^-} f = \lim_{x\to a^+} f = L$;
• identify when a limit fails to exist (oscillation, one-sided limits disagree, unbounded growth).

1. Two motivating problems
2. The informal definition
3. One-sided limits
4. When limits fail to exist