How to compare two functions at infinity for asymptotic analysis? On this wikipedia page its mentioned that "asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity"
But on this page it has compared $n^2+3n$ and $n^2$ and claimed them to be asymptotically equivalent.
I get it that their ratio approaches one as $x \to \infty$ which means they are equal.
But their difference is $3n$ which is also approaching $\infty$. How can two functions with infinite difference be equal? Also it goes against the first links definition that the distance should be approaching zero, which is not the case here as its approaching $\infty$
The basic problem which I am facing is (according to me) : Are $1000000100000$ and $1000000000000$ nearly equal? Their ratio is pretty close to one but there difference is 100000, which is quite big I think.
 A: There are two distinct notions at play here. For a given function $f$, the first notion is an asymptote, a line $l$ whose distance $|l(x)-f(x)|$ from $f$ tends to $0$ as $x\to\infty$, while the second notion is asymptotic equivalence/equality, a relation satisfied by a function $g$ whose quotient $\frac{g(x)}{f(x)}$ with $f$ tends to $1$ as $x\to\infty$. The first represents a line, specifically, that $f$ gets near to, in the ordinary, additive sense of "near" as a small absolute difference. But the latter represents a function that grows or vanishes at a similar rate to $f$, in a multiplicative sense. This might immediately ring alarm bells that they aren't the same thing, since a line has only one rate of growth: linear!
Now, either difference near $0$ or a quotient near $1$ can represent some sense of nearness. But one does not necessarily imply the other. We can refer to these two additive and multiplicative types of nearness as absolute and relative, respectively. For example, considering only numbers, $0.001$ and $0.000\,001$ are both tiny and hence near one another in an absolute sense, but they are far in a relative sense since one is $1000$ times the other. And vice versa, $100\,000\,000$ and $100\,010\,000$ may be absolutely far as they have a difference of $10\,000$, but as they are huge numbers on the same order (i.e., number of digits), they are relatively close!
Translating this idea back to functions, as $x\to\infty$, both $1/x$ and $1/x^2$ tend to $0$ but at different rates. Hence, they tend nearer to one another absolutely but not relatively (i.e., they are not asymptotic). On the other hand, $x$ and $x+1$ maintain a constant difference that is dwarfed by their size. Hence, they tend nearer to one another relatively (i.e., they are asymptotic) but not absolutely.
So, an "asymptote (line) of $f$" may not actually be "asymptotic to $f$". It should generally be obvious from context what is meant, but be careful about your terminology.
