Before we can differentiate or integrate anything, we need a precise way to talk about what a function approaches. A limit answers the question: as the input slides toward some value , does the output settle near a single number ? Crucially, the limit ignores what happens at itself; it cares only about the neighborhood around .
Knowledge check
Q1. In the epsilon-delta definition, what is the logical order of quantifiers?
Q2. Why does the two-sided limit of x/|x| fail to exist at 0?
Q3. The function (x^2-1)/(x-1) at x=1 has which kind of discontinuity?
Q4. The Intermediate Value Theorem guarantees a root of x^3 - x - 1 in (1,2) because:
Problems
Compute the limit as x approaches infinity of (3x^2 + 5)/(x^2 - 1). Give a decimal.
Using the epsilon-delta definition for lim_{x->3}(2x+1)=7, find the largest delta that works for epsilon = 0.1. Give a decimal.
This distinction matters for quants. A bond price as time-to-maturity shrinks, an option's implied volatility as the strike approaches the forward, a Monte Carlo estimator as the path count grows: each is a limit, and each may behave badly exactly at the point of interest while being perfectly well-behaved nearby.
The rigorous definition pins down "approaches" with two tolerances.
Read it as a game: you name an output tolerance , and I must produce an input tolerance so that every within of (but not equal to ) lands within of . If I can always win, the limit is .
Claim: . Given , we need so that forces . Now
We want , i.e. . So choose . Then gives . The game is won for every , so the limit is exactly .
Sometimes the approach direction matters. The right-hand limit restricts to ; the left-hand limit restricts to . The two-sided limit exists if and only if both one-sided limits exist and agree.
Example: for , the right limit at is and the left limit is , so does not exist.
A limit at infinity describes long-run behavior: means can be forced within of for all sufficiently large . Formally, for every there is an such that implies ; the threshold plays the role played before. For instance
found by dividing numerator and denominator by and noting the and terms vanish. The same technique handles the asymptotic yield of a perpetuity or the limiting price of a long-dated cash flow.
Limits also obey clean algebraic laws that let us avoid epsilon-delta for routine work. If and both exist, then the limit of a sum, difference, product, or quotient (denominator limit nonzero) is the corresponding combination of the limits. These laws, provable from the definition, are why we can compute most limits by substitution once continuity is established.
A function is continuous at when the limit exists, the value exists, and they coincide:
Three standard ways this breaks:
If is continuous on a closed interval and lies between and , then there exists in with . Concretely, take . Since and , continuity guarantees a root somewhere in . This is the theoretical engine behind bisection root-finding.
Continuity is the license to use calculus at all. Pricing functions must be continuous in their inputs for hedge ratios (derivatives) to exist and for interpolation of yield curves to be sound. The IVT underpins bisection, the workhorse for inverting Black-Scholes to recover implied volatility from a market price: because price is continuous and monotone in volatility, a bracketed root must exist and can be squeezed numerically. Jump discontinuities flag genuine modeling hazards: a payoff with a jump (a digital option) cannot be delta-hedged smoothly near the strike, which is exactly where risk concentrates.
Adapted from MIT OpenCourseWare (18.01 / 18.02 Calculus), CC BY-NC-SA.