Limits

Where this is used

A machine learning model trains in a loop that keeps going "until the loss settles down toward 0" — that is a limit. The continuously compounded interest a bank quotes comes from the very same limit that defines e. And the laws of quantum mechanics, fluid dynamics, and statistical mechanics simply cannot be written down without limits. Derivatives and integrals are both built on top of them.

Start gently

A limit asks where something is heading as it gets closer and closer. It need never arrive; only the destination matters.

Key points

The limit of a sequence

If the terms of a sequence {a_n} get closer and closer to a single value L as n grows without bound, we write lim_{n→∞} a_n = L and say the sequence converges to L.

Limits of geometric sequences

When |r| < 1, lim_{n→∞} rⁿ = 0. When |r| > 1 the terms blow up; when r = 1 they stay at 1; and when r = −1 they flip back and forth forever.

Infinite geometric series

For a first term a and a common ratio r with |r| < 1, the endless sum comes to S = a/(1−r). This only works — the series only converges — when |r| < 1.

The number e

e = lim_{n→∞}(1+1/n)^n ≈ 2.71828… . It is one of the most important constants in mathematics, sitting underneath continuous compounding, exponential functions, and calculus.

How limits give you derivatives

The derivative is defined as f'(x) = lim_{h→0} [f(x+h)−f(x)]/h. Without the limit as h → 0, there is no way to define a derivative at all.

See it drawn

(1 + 1/n)ⁿ approaches e ≈ 2.718

It keeps rising, yet never passes 2.718…. That ceiling is e, the base of the natural logarithm.

Jobs that use this

Machine Learning Engineer$180k

Quantitative Analyst (Quant)$250k

Theoretical Physicist$130k

Start gently

Key points

See it drawn

Jobs that use this

Members-only from here

Comments