Relative Rates of Growth

Definition

Relative rates of growth allow us to compare the speed at which functions approach infinity as $x \to \infty$.

• $f$ grows faster than $g$ if $\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$.

• How to read: “The limit as x approaches infinity of f of x divided by g of x equals infinity.”

  • Meaning: $f$ eventually dominates $g$—the ratio blows up, so $f$ outpaces $g$ at infinity.

• $f$ and $g$ grow at the same rate if $\lim_{x \to \infty} \frac{f(x)}{g(x)} = L$ ($0 < L < \infty$).

• How to read: “The limit equals L, where L is a positive finite number.”

  • Meaning: $f$ and $g$ are asymptotically proportional—they differ only by a constant factor at infinity.

Why It Matters

In a world of finite resources, speed is relative. If your costs grow faster than your revenue, you are bankrupt regardless of current volume. Asymptotic analysis is the only way to predict the long-term viability of algorithms and businesses.

Core Concepts

• Hierarchy of Growth: Logarithms grow slower than polynomials, which grow slower than exponentials, which grow slower than factorial-like functions ($x^x$).

• How to read: “The X to the x.”

  • Meaning: The growth hierarchy: $\ln x \ll x^n \ll a^x \ll x^x$ for large $x$.

• Dominance: In a sum of functions, the fastest-growing term eventually dominates the behavior of the entire expression.

• Asymptotic Analysis: Focuses on long-term behavior rather than specific values.

Connected Concepts

• Exponential Growth Model
• Exponential Decay Model
• Rates of Change in Scientific Applications
• Related Rates