Limits

Motivation: why this matters in quant finance

The limit is the foundational concept of calculus. Every derivative, every integral, every convergence result is defined through a limit. In quantitative finance, limits appear in at least three distinct guises:

Defining the derivative. The sensitivity of an option price to the underlying — the delta $\Delta = \partial v / \partial S$ — is the limit of a difference quotient $\Delta v / \Delta S$ as $\Delta S \to 0$ . Without a rigorous notion of limit, the entire differentiation framework and hence the Greeks would be undefined.
Passage from discrete to continuous time. Much of quantitative finance involves taking a discrete model and letting the time step $\Delta t \to 0$ . The binomial tree model converges to geometric Brownian motion; a discrete random walk converges (via Donsker's theorem) to Brownian motion. These are all limit statements.
Convergence of sums and integrals. Quadratic variation — the fact that $\sum (W_{t_{i+1}} - W_{t_i})^2 \to T$ as the partition refines — is a limit in $L^2$ and in probability. The entire construction of the Itô integral is a limit of sums of simple processes. Without understanding limits, these convergence results are opaque.

In short, limits are the invisible machinery behind virtually every formula in continuous-time finance. This page defines them precisely, builds intuition, and connects them to where they appear in the quant-finance curriculum.

The informal idea

The statement "

\lim_{x \to a} f(x) = L

" means: as

x

gets closer and closer to

a

(without necessarily equalling

a

f(x)

gets closer and closer to

L

. The value of

f

a

itself is irrelevant —

f(a)

may not even be defined. What matters is the behaviour near

a

Example: Let

f(x) = \frac{x^2 - 1}{x - 1}

. This is undefined at

x = 1

(division by zero). But for

x \neq 1

, we can factor:

f(x) = \frac{(x-1)(x+1)}{x-1} = x + 1

. So as

x \to 1

f(x) \to 2

. We write

\lim_{x \to 1} f(x) = 2

, even though

f(1)

does not exist.

This pattern — a quantity that is undefined or indeterminate at a point but has a well-defined limiting value — is exactly what happens with the derivative: $\frac{f(x + h) - f(x)}{h}$ is undefined at $h = 0$ , but its limit as $h \to 0$ is the derivative $f'(x)$ .

The formal definition: epsilon-delta

Finite limits at finite points

The rigorous definition, due to Weierstrass, eliminates the vagueness of "closer and closer":

$\lim_{x \to a} f(x) = L$ if and only if: for every $\varepsilon > 0$ , there exists a $\delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$

In words: no matter how tight a tolerance $\varepsilon$ you specify around $L$ , I can find a neighbourhood of radius $\delta$ around $a$ such that every $x$ in that neighbourhood (except possibly $a$ itself) maps to within $\varepsilon$ of $L$ .

The key features:

$\varepsilon$ comes first. The definition is "for all $\varepsilon$ , there exists $\delta$ " — the challenger picks the tolerance, and then you must respond with a neighbourhood. This order matters and is the source of the definition's logical power.
$0 < |x - a|$ excludes $x = a$ itself. The limit is about behaviour near $a$ , not at $a$ .
$\delta$ may depend on $\varepsilon$ (and typically does). Usually, a tighter tolerance requires a smaller neighbourhood.

Worked example: proving $\lim_{x \to 3} (2x + 1) = 7$

We need: for every $\varepsilon > 0$ , find $\delta > 0$ such that $0 < |x - 3| < \delta \implies |(2x + 1) - 7| < \varepsilon$ .

Simplify the consequent: $|2x + 1 - 7| = |2x - 6| = 2|x - 3|$ .

So we need $2|x - 3| < \varepsilon$ , i.e., $|x - 3| < \varepsilon / 2$ .

Choose $\delta = \varepsilon / 2$ . Then $0 < |x - 3| < \delta$ implies $|f(x) - 7| = 2|x - 3| < 2\delta = \varepsilon$ . $\square$

This is a simple case because $f$ is linear. For nonlinear functions, finding $\delta$ requires more work, but the logical structure is always the same.

Limits at infinity

\lim_{x \to \infty} f(x) = L \iff \forall\,\varepsilon > 0,\;\exists\,M > 0: x > M \implies |f(x) - L| < \varepsilon

Finance example: Consider a zero-coupon bond price

P(T) = e^{-rT}

as a function of maturity. As

T \to \infty

P(T) \to 0

: the present value of a dollar received infinitely far in the future is zero (assuming

r > 0

). Formally, for any

\varepsilon > 0

, choose

M = -\ln(\varepsilon)/r

; then

T > M

implies

e^{-rT} < \varepsilon

Infinite limits

\lim_{x \to a} f(x) = \infty \iff \forall\,M > 0,\;\exists\,\delta > 0: 0 < |x - a| < \delta \implies f(x) > M

Finance example: As noted in Brownian Motion, the "derivative"

\Delta W / \Delta t

of a Brownian path diverges:

|\Delta W / \Delta t| \sim 1/\sqrt{\Delta t} \to \infty

\Delta t \to 0

. This is an infinite limit that shows the derivative does not exist — the difference quotient grows without bound instead of converging to a finite value.

One-sided limits

Sometimes the behaviour of

f

approaching from the left differs from approaching from the right. The left-hand limit and right-hand limit are:

\lim_{x \to a^-} f(x) = L^- \qquad \text{(approach from below)}

\lim_{x \to a^+} f(x) = L^+ \qquad \text{(approach from above)}

The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal: $L^- = L^+$ .

Finance example: The call option payoff

H(S) = (S - K)^+

at expiration. At

S = K

\lim_{S \to K^-} H(S) = 0, \qquad \lim_{S \to K^+} H(S) = 0

Both one-sided limits equal zero, and

H(K) = 0

, so

H

is continuous at

K

. But the derivative behaves differently from each side:

\lim_{S \to K^-} H'(S) = 0, \qquad \lim_{S \to K^+} H'(S) = 1

The one-sided limits of the derivative disagree, so $H'(K)$ does not exist — this is the kink in the payoff function. The delta of a call at expiration jumps from 0 to 1 at the strike, which is why near-expiry ATM options have enormous gamma.

Limit laws

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ , then:

Law	Statement
Sum	$\lim (f + g) = L + M$
Scalar multiple	$\lim (cf) = cL$
Product	$\lim (f \cdot g) = L \cdot M$
Quotient	$\lim (f / g) = L / M$ , provided $M \neq 0$
Power	$\lim f^n = L^n$
Composition	$\lim f(g(x)) = f(M)$ if $f$ is continuous at $M$

These laws are what make calculus computationally tractable: you can break a complicated limit into simpler pieces. The composition law is particularly important — it is the rigorous foundation of the chain rule.

Linearity (sum + scalar multiple) is the reason that limits distribute over portfolio operations: if each asset's price converges, the portfolio value converges, and the limit of the portfolio is the portfolio of the limits. This is used implicitly every time we write

\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]

, which is itself a limit statement (the expectation is defined as an integral, which is a limit of sums).

Indeterminate forms and L'Hôpital's rule

When direct substitution into a limit gives an expression like

0/0

\infty/\infty

, the limit is called an indeterminate form. The most common indeterminate forms are:

\frac{0}{0}, \quad \frac{\infty}{\infty}, \quad 0 \cdot \infty, \quad \infty - \infty, \quad 0^0, \quad 1^\infty, \quad \infty^0

L'Hôpital's rule resolves the first two: if

\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0

(or both

\to \pm\infty

), and

g'(x) \neq 0

near

a

, then:

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

provided the right-hand limit exists (or is $\pm\infty$ ).

Worked example: continuous compounding as a limit

The definition of $e$ (and hence continuous compounding) is a classic limit:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

More generally, the future value of $1 invested at annual rate $r$ , compounded $n$ times per year for time $T$ , is:

\text{FV} = \left(1 + \frac{r}{n}\right)^{nT}

Taking the limit as $n \to \infty$ :

\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nT} = e^{rT}

This is a $1^\infty$ indeterminate form. To evaluate it, take logarithms:

\ln\text{FV} = nT \ln\left(1 + \frac{r}{n}\right)

As $n \to \infty$ , this is an $\infty \cdot 0$ form. Rewrite as a $0/0$ form and apply L'Hôpital's rule with $u = 1/n \to 0$ :

\lim_{u \to 0} \frac{T\ln(1 + ru)}{u} = T \lim_{u \to 0} \frac{r/(1 + ru)}{1} = rT

\ln\text{FV} \to rT

, hence

\text{FV} \to e^{rT}

. This is why the discount factor

e^{-rT}

appears throughout quant finance — it is the continuous-compounding limit. See Discounting for the financial context.

Worked example: the derivative of $e^x$ from the limit definition

\frac{d}{dx}e^x = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} = e^x \lim_{h \to 0} \frac{e^h - 1}{h}

The remaining limit is a $0/0$ indeterminate form. By L'Hôpital's rule (or by the series expansion $e^h = 1 + h + h^2/2 + \cdots$ ):

\lim_{h \to 0} \frac{e^h - 1}{h} = 1

\frac{d}{dx}e^x = e^x

, confirming the exponential rule from the differentiation introduction.

Limits of sequences and series

Sequences

A sequence

\{a_n\}_{n=1}^\infty

converges to

L

if:

\forall\,\varepsilon > 0,\;\exists\,N \in \mathbb{N}: n > N \implies |a_n - L| < \varepsilon

This is the discrete analogue of the limit at infinity. In quant finance, convergence of sequences appears when:

Binomial tree prices converge to Black-Scholes prices as the number of steps $n \to \infty$ .
Monte Carlo estimators converge to the true expected value by the law of large numbers.
Quadratic variation sums $Q_n = \sum_{i}(\Delta W_i)^2$ converge to $T$ as the partition refines — a key fact in Brownian motion.

Series

A series

\sum_{n=0}^\infty a_n

is the limit of the partial sums

S_N = \sum_{n=0}^N a_n

\sum_{n=0}^\infty a_n = \lim_{N \to \infty} S_N

The most important series in this vault is the Taylor series:

f(x + h) = \sum_{n=0}^\infty \frac{f^{(n)}(x)}{n!}h^n

which converges (for analytic functions) when

|h|

is small enough. The Taylor series is the bridge between the limit-based definition of the derivative and the Itô expansion, which is a Taylor series truncated at second order because

(dW)^2 = dt

makes the second-order term survive while higher orders vanish.

Modes of convergence in probability (preview)

In stochastic calculus, "limit" can mean several different things depending on the mode of convergence. This is a brief preview; the details belong in the probability and measure theory pages.

Almost sure convergence:

X_n \to X

a.s. means

\mathbb{P}(\lim_{n \to \infty} X_n = X) = 1

. The strongest mode — the sequence converges for (almost) every outcome

\omega

Convergence in probability:

X_n \xrightarrow{p} X

means

\mathbb{P}(|X_n - X| > \varepsilon) \to 0

for every

\varepsilon > 0

. Weaker than a.s. convergence but still useful.

Convergence in $L^p$ :

X_n \xrightarrow{L^p} X

means

\mathbb{E}[|X_n - X|^p] \to 0

. The Itô integral is constructed as an

L^2

limit of sums, meaning the expected squared error between the sum and the integral tends to zero.

Convergence in distribution:

X_n \xrightarrow{d} X

means

\mathbb{P}(X_n \le x) \to \mathbb{P}(X \le x)

at all continuity points of the CDF. Donsker's theorem (the random walk scaling to Brownian motion) is a convergence in distribution result.

Each mode is a different kind of limit, and knowing which one applies is essential for understanding the rigour of results like quadratic variation, Itô integration, and the central limit theorem.

Common confusions and pitfalls

Confusing the limit with the function value.

\lim_{x \to a} f(x)

may differ from

f(a)

, or

f(a)

may not exist at all. The limit is about the neighbourhood of

a

, not the point itself. When these do agree —

\lim_{x \to a} f(x) = f(a)

— we say

f

is continuous at

a

Assuming limits always exist. They don't. The limit

\lim_{x \to 0} \sin(1/x)

does not exist because the function oscillates between

-1

and

1

without settling. In finance, a model whose output oscillates wildly with small parameter changes may not have well-defined sensitivities — this is a numerical (and conceptual) problem.

Using L'Hôpital's rule when it doesn't apply. L'Hôpital's rule requires an indeterminate form (

0/0

\infty/\infty

). Applying it to a non-indeterminate limit like

\lim_{x \to 1} (x+1)/x = 2/1 = 2

gives a wrong answer (

1/1 = 1

). Always check the form before applying the rule.

Confusing convergence modes in probability. "The sum converges" is ambiguous without specifying the mode. Quadratic variation converges in

L^2

(and in probability); the Itô integral converges in

L^2

; Donsker's theorem is convergence in distribution. These are different statements with different implications, and conflating them leads to false conclusions.

Where this goes next

The limit is the gateway to the two pillars of calculus:

Continuity — the condition that $\lim_{x \to a} f(x) = f(a)$ . Continuity is what makes Brownian motion paths well-behaved enough to do stochastic calculus on, and its failure (discontinuities / jumps) is what motivates jump-diffusion extensions.
Differentiation — the limit of the difference quotient. All of the chain rule, product rule, and quotient rule are derived from limit manipulations, and Itô's Lemma is the stochastic extension of these limit-based rules.

The Taylor series is the most important application of limits to the quant-finance curriculum: it approximates smooth functions by polynomials and is the starting point for the Itô expansion that leads to the Black-Scholes PDE.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 2 Sections 2.2-2.4 (limits, limit laws, and the precise definition of a limit).

Limits

Motivation: why this matters in quant finance

The informal idea

The formal definition: epsilon-delta

Finite limits at finite points

Worked example: proving lim⁡x→3(2x+1)=7\lim_{x \to 3} (2x + 1) = 7limx→3​(2x+1)=7

Limits at infinity

Infinite limits

One-sided limits

Limit laws

Indeterminate forms and L'Hôpital's rule

Worked example: continuous compounding as a limit

Worked example: the derivative of exe^xex from the limit definition

Limits of sequences and series

Sequences

Series

Modes of convergence in probability (preview)

Common confusions and pitfalls

Where this goes next

References

Worked example: proving $\lim_{x \to 3} (2x + 1) = 7$

Worked example: the derivative of $e^x$ from the limit definition