Two functions are asymptotically equivalent near a point when their ratio approaches $1$. We write

$$f(x)\sim g(x)\qquad(x\to a)$$

when

$$\lim_{x\to a}\frac{f(x)}{g(x)}=1.$$

The notation means that, close to the limiting point, the two functions have the same leading behavior.

A simple example

Near $x=0$,

$$\sin x\sim x$$

because

$$\lim_{x\to0}\frac{\sin x}{x}=1.$$

Thus, in a quotient such as

$$\frac{\sin x}{x},$$

replacing $\sin x$ by its equivalent $x$ immediately reveals the limit $1$.

Useful equivalences near zero

Several common functions have simple first-order behavior:

$$\sin x\sim x,$$

$$e^x-1\sim x,$$

$$\ln(1+x)\sim x.$$

For example,

$$\frac{e^x-1}{\sin x}\sim\frac{x}{x}=1.$$

Where replacement is valid

Asymptotic equivalents can be safely substituted as factors in products and quotients when the resulting expressions are defined. They cannot in general be substituted term by term inside sums or differences, because cancellation may remove the leading behavior.

The purpose of an asymptotic equivalence is not to claim that two functions are equal, but that their ratio becomes indistinguishable from $1$ in the specified limit.