Approximaths

Month: November 2026

  • Asymptotics: Introduction I

    We say that f is asymptotic to g as x \to x_0 and write:

    f(x) \sim g(x) \quad \text{if} \quad \lim_{x \to x_0} \frac{f(x)}{g(x)} = 1

    Typically, f is a “complicated” function and g is a “simpler” function.

    Here are a few simple examples:

    • \sin(x) \sim x \quad \text{as } x \to 0 \quad \text{since} \quad \lim_{x \to 0}\frac{\sin(x)}{x}=1
    • e^x-1 \sim x \quad \text{as } x \to 0 \quad \text{since} \quad \lim_{x \to 0}\frac{e^x-1}{x}=1
    • e^x \sim 1 \quad \text{as } x \to 0 \quad \text{since} \quad \lim_{x \to 0}\frac{e^x}{1}=1

    In the vicinity of x_0, we have replaced a “complicated” function with a simpler one (see Table 1).

    Complicated Simpler x_0
    \sin(x) x 0
    e^x-1 x 0
    e^x 1 0

    Using asymptotics, we have replaced a complex function in the vicinity of x0 by a simpler function.