Asymptotics: Introduction I

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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:

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