Additively asymptotic functions

Definition

Suppose $f$ and $g$ are both functions whose domain contains an interval of the form $(a,\infty )$ for some real number $a$. We say that two functions $f$ and $g$ are additively asymptotic in the $+\infty$-direction if:

$\lim _{x\to \infty }(f(x)-g(x))=0$

We say that $f$ and $g$ are additively asymptotic in the $-\infty$-direction if:

$\lim _{x\to -\infty }(f(x)-g(x))=0$

Usually, when we just say that $f$ and $g$ are additively asymptotic, we need to infer from context whether this refers to the $+\infty$-direction, the $-\infty$-direction, or both directions.

Caveat

Note that the statement:

$\lim _{x\to \infty }(f(x)-g(x))=0$

is not the same as the statement:

$\lim _{x\to \infty }f(x)=\lim _{x\to \infty }g(x)$

The two statements are equivalent if either of the limits are finite. However, if both limits are infinite, the fact that they are equal (i.e., they are both $+\infty$ or both $-\infty$) is a much weaker condition than saying that the difference approaches zero. The key issue is the rate at which the functions approach infinity. Thus, even though two functions both approach infinity, their difference need not approach zero.