Why can we replace an infinitesimal in a limit with an equivalent infinitesimal?

I read the following in a website.

I want to know why we can replace one infinitesimal with an equivalent one. The idea seems intuitive but is there a formal proof?

Answer

So basically it's because for equivalent infinitesimal expressions of a function, the limit of its ratio with the original function as x approach 0 can be proved to be 1. (sin x/x for example. Sorry, I'm typing on a phone so the format is crude.)

Source: Wikipedia - Indeterminate Form.