Limits

A limit describes the value a function or sequence approaches as its input moves toward some point. Biology is full of such limiting behaviors: as an epidemic runs its course the susceptible fraction approaches a fixed limit — the final epidemic size — and long-run averages of case counts converge. Limits are the foundation of derivatives, integrals, and the convergence theorems that make statistical inference work.

Intuitive and formal definition#

Intuitively, $\lim_{n\to\infty} a_n = L$ means the terms $a_n$ get and stay arbitrarily close to $L$ . Formally ( $\epsilon$ -definition): for every $\epsilon > 0$ there exists an $N$ such that

$n > N \implies |a_n - L| < \epsilon.$

If such an $L$ exists the sequence converges; otherwise it diverges.

Properties of limits#

If $\lim a_n = A$ and $\lim b_n = B$ , then

$\lim (a_n + b_n) = A + B,\quad \lim (a_n b_n) = AB,\quad \lim \frac{a_n}{b_n} = \frac{A}{B}\ (B \neq 0).$

L’Hôpital’s rule#

For an indeterminate form $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ ,

$\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}.$

Example: $\displaystyle \lim_{x\to 0}\frac{\sin x}{x}$ is $\tfrac{0}{0}$ . Differentiating top and bottom gives $\lim_{x\to 0}\frac{\cos x}{1} = \cos 0 = 1$ . The same $\tfrac{0}{0}$ manoeuvre appears in disease models when taking small-time or large-population approximations, such as recovering a per-capita infection rate as a time interval shrinks to zero.

Worked example#

Consider $a_n = \frac{3n + 1}{n}= 3 + \frac{1}{n}$ . As $n$ grows, $1/n \to 0$ , so $a_n \to 3$ . Checking $\epsilon = 0.01$ : we need $|a_n - 3| = 1/n < 0.01$ , i.e. $n > 100$ .

Computing it#

R#

# Numerically approach lim_{x->0} sin(x)/x
x <- 10^(-(1:6))
sin(x) / x        # 0.9983..., -> 1

Python#

Python

import sympy as sp

x, n = sp.symbols("x n")
print(sp.limit(sp.sin(x)/x, x, 0))        # 1
print(sp.limit((3*n + 1)/n, n, sp.oo))    # 3

1
3

Julia#

Julia

using Symbolics

@variables x
# Numeric check of sin(x)/x near 0
xs = 10.0 .^ (-(1:6))
sin.(xs) ./ xs        # -> 1.0

Why it matters for statistics#

Convergence of sequences of random quantities is the engine of large-sample theory. The Weak Law of Large Numbers says the sample mean converges in probability to the true mean, $\bar{X}_n \xrightarrow{p} \mu$ — a probabilistic limit. Understanding ordinary limits first makes these stochastic versions far less mysterious.