[Math] Why can we replace an infinitesimal in a limit with an equivalent infinitesimal

Question

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 1

Note that in general you cannot simply replace arbitrary quantities by infinitesimally equivalent ones. For example, $\lim_{x \to 0} \frac{\sin(x)-x}{x^3}$ is not zero, which is what you would get if you replaced $\sin(x)$ by $x$ in it. In most cases what you are doing when you replace infinitesimally equivalent quantities is multiplying the final result by $1$, writing $1$ as a limit of a ratio of infinitesimally equivalent quantities, and then dragging this limit inside your original one. So for example in the first problem in the image, the steps look like:

$$\lim_{x \to 0} \frac{\ln(1+4x)}{\sin(3x)}=\lim_{x \to 0} \frac{\ln(1+4x)}{\sin(3x)} \lim_{x \to 0} \frac{4x}{\ln(1+4x)} \lim_{x \to 0} \frac{\sin(3x)}{3x}=\lim_{x \to 0} \frac{4x}{3x}=4/3.$$