# BOB0 - BOTN - EATS DC

BOB0 - BOTN - EATS DC is one of those horrible mnemonic devices that pride-less students love. (Other such monstrosities include cross multiplication, FOIL, SOH-CAH-TOA, and that stupid quadratic formula song.) Let's unpack this mnemonic, I guess. I am not excited about this, but, also being pride-less, I will reluctantly do this if it means you will understand non-vertical asymptotes of rational functions.

## "BOB0" means "Bigger On Bottom: 0"

Or, with slightly more detail, if a rational function is "Bigger On Bottom", then it has a horizontal aymptote of . Or, if you want to say it in a (prideful) way that won't make me cringe:
Let be a rational function that we write as , where and are polynomials, and let the degree of be greater than the degree of . Then has a horizontal asymptote of .

## Plot a "BOB0" rational function.

Let be a BOB0 rational function, so therefore we can write a limit to describe its end behavior. We can write , where...

Which of the following could be ?

Which of the following could be ?

Explain in your own words why BOB0 works. (Meaning, why is it the case that Bigger On Bottom results in a horizontal asymptote of ?)

## "BOTN" means...

Well, what do you think BOTN is meant to convey? Try to explain it in a way that won't make me cringe. If you need to experiment before you answer, skip ahead to the next question.

## Plot a "BOTN" rational function.

Let be a BOTN rational function, so therefore we can write a limit to describe its end behavior. We can write , where...

Which of the following could be ?

Which of the following could be ?

Explain in your own words why BOTN works.

## "EATS DC" means "Exponents Are The Same: Divide Coefficients"

Okay, so the way I would articulate EATS DC this way:
Let be a rational function where is a polynomial with leading coefficient , and is a polynomial with leading coefficient . Let the degree of be equal the degree of . Then has a horizontal asymptote of .

## Plot an "EATS DC" rational function.

Let be an EATS DC rational function where is a polynomial with leading coefficient , and is a polynomial with leading coefficient . We can therefore write to describe 's end behavior, where...

Which of the following could be ?

Which of the following could be ?