Does the rule " divide by highest denominator power" apply outside the field of rational functions? ( Context : limits and asymptotes)

Note : I assume here that the function $f(x)$ as defined below is not a rational function ( i.e. a quotient of polynomials).

I found this question in a " maturité" test ( Swiss high school final exam), dating from $2014$ :

Let $\LARGE f $ be a function defined by : $\LARGE f(x)= \frac {x^2 - e^{x+1}} {x^2 +e^x}$ . Justfy that $\mathbb R$ is the domain of $\LARGE f$ and determine the asymptotes of function $\LARGE f$.

In order to find a possible horizontal asymptote, I asked Symbolab :

$\LARGE lim_{x\rightarrow\infty} \bigg(\frac {x^2 - e^{x+1}} {x^2 +e^x}\bigg)$.

I observe that, in order to tackle this problem, Symbolab applies a rule that , according to what I thought , only held for rational functions, namely the rule :

" divide by the highest denominator power",

whch yields ( with $\LARGE e^x$ playing the role of " highest denominator power ") :

$\LARGE \frac {\frac {x^2}{e^x} - e^1} {\frac {x^2}{e^x}+ 1}$

I have two questions :

(1) does the rule " divide by highest denominator power" actually apply outside the field of rational functions ( i.e. quotients of polynomials)?

(2) since $x$ is a variable that may ( apparently) take values less that two, what justfies the fact of identifying the " highest denominator power " to the second term, namely $\LARGE e^x$ and not to $\LARGE x^2$?

Exam :

Solution 1:

1. To answer your first question, we want to divide by the largest term in the denominator (as a generalization of the highest denominator power rule for rational functions).

Since $x\rightarrow\infty$, we want to divide by the largest quantity then, and in that case, it is indeed the exponential term.

This answer explains why the exponential function grows faster than any polynomial function, given sufficiently large values of $x$.

2. Your second question does not seem relevant for this question since we want $x\rightarrow\infty$, not $x$ near 2.

Solution 2:

Actually, what is being done is not “divide by the highest denominator power”; it is “divide by the thing that grows faster”. In this case, that thing is $e^x$ ($e^{x+1}$ is another possibility, since it grows at the same rate as $e^x$). And, yes, it's a good approach.