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- How to interpret the phrase "The function f(x) tends to the function g(x) when x tends to x0"

Hi, I have a question, sometimes one sees in exercises or textbooks some phrase like

$$\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)$$

Another way (I think) this can be interpreted is that

$$\lim_{x\to x_0} \frac{f(x)}{g(x)}=1$$

But also, in addition to ##\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)## one can think also to include a more restrictive condition

$$\lim_{x\to x_0} f'(x)=\lim_{x\to x_0} g'(x)$$

And so on..., even you could try to define that both function must have the same Taylor series arround ##x_0## etc...

So is there a unambiguos definition of that phrase or simply something that physicist say that depend completely on the point of view of the person?

My question is, is this unambiguously defined in a mathematical way? I mean, when one reads such a thing this could mean thatThe function ##f(x)## tends to the function ##g(x)## when x tends to ##x_0##

$$\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)$$

Another way (I think) this can be interpreted is that

$$\lim_{x\to x_0} \frac{f(x)}{g(x)}=1$$

But also, in addition to ##\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)## one can think also to include a more restrictive condition

$$\lim_{x\to x_0} f'(x)=\lim_{x\to x_0} g'(x)$$

And so on..., even you could try to define that both function must have the same Taylor series arround ##x_0## etc...

So is there a unambiguos definition of that phrase or simply something that physicist say that depend completely on the point of view of the person?