The three fundamental theorems

Introduction

When studying a limit at +∞ involving several types of functions (logarithmic, power, exponential), it is not enough to analyse each term individually: one must understand the hierarchy of growth between these families of functions. This is the subject of the comparative growth theorems.

The three fundamental results

For any integer n ≥ 1 (and indeed any real number n > 0):

1. lim x → +∞ (ln(x))/x^n = 0
2. lim x → +∞ x^n / e^x = 0
3. lim x → 0⁺ x^n * ln(x) = 0

Hierarchy to remember

Function	Rate of growth
ln(x)	very slow
x^n (n > 0)	moderate
e^x	very fast

This is often summarised as: ln(x) << x^n << e^x as x approaches +∞ (the symbol << means ‘negligible compared to’).

Examples

• lim x → +∞ (ln x)/√x = 0, because √x = x^(1/2) dominates ln(x).
• lim x → +∞ x⁵ * e^(-x) = lim x → +∞ x⁵/e^x = 0.
• lim x → 0⁺ x² * ln(x) = 0 despite the fact that ln(x) → -∞.

Common pitfall

Do not confuse comparative growth with order of magnitude for a fixed x. These theorems apply to limits as x approaches +∞ (or 0+ for the third one), not as x approaches -∞ nor for a particular value of x.