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.
Or, with slightly more detail, if a rational function is "Bigger On Bottom", then it has a horizontal aymptote of [math]y=0[/math]. Or, if you want to say it in a (prideful) way that won't make me cringe:[br][br]Let [math]r[/math] be a rational function that we write as [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math], where [math]p[/math] and [math]q[/math] are polynomials, and let the degree of [math]q[/math] be greater than the degree of [math]p[/math]. Then [math]r[/math] has a horizontal asymptote of [math]y=0[/math].[br][br]
Let [math]f(x)[/math] be a BOB0 rational function, so therefore we can write a limit to describe its end behavior. We can write [math]\lim_{x\to A}f\left(x\right)=B[/math], where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?
Explain in your own words why BOB0 works. (Meaning, why is it the case that Bigger On Bottom results in a horizontal asymptote of [math]y=0[/math]?)
If the degree of [math]q[/math] is greater than the degree of [math]p[/math], then [math]\left|q\left(x\right)\right|>>\left|p\left(x\right)\right|[/math] for [math]x[/math]-values that are very positive or very negative. That is, sufficiently far to the left of (and sufficiently far to the right of) [math]x=0[/math], [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] is close to (but not equal to) [math]0[/math].
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.
Let [math]r[/math] be a rational function that we write as [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math], where [math]p[/math] and [math]q[/math] are polynomials, and let the degree of [math]p[/math] be greater than the degree of [math]q[/math]. Then [math]r[/math] has no horizontal asymptote.
Let [math]f(x)[/math] be a BOTN rational function, so therefore we can write a limit to describe its end behavior. We can write [math]\lim_{x\to A}f\left(x\right)=B[/math], where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?
Explain in your own words why BOTN works.
If the degree of [math]p[/math] is greater than the degree of [math]q[/math], then [math]\left|p\left(x\right)\right|>>\left|q\left(x\right)\right|[/math] for [math]x[/math]-values that are very positive or very negative. That is, sufficiently far to the left of (and sufficiently far to the right of) [math]x=0[/math], [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] is extremely large (which could mean very positive or very negative). Moving even further left (or further right) of that will make [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] larger yet.
Okay, so the way I would articulate EATS DC this way:[br][br]Let [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] be a rational function where [math]p[/math] is a polynomial with leading coefficient [math]a[/math], and [math]q[/math] is a polynomial with leading coefficient [math]b[/math]. Let the degree of [math]p[/math] be equal the degree of [math]q[/math]. Then [math]r[/math] has a horizontal asymptote of [math]y=\frac{a}{b}[/math].
Let [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] be an EATS DC rational function where [math]p[/math] is a polynomial with leading coefficient [math]a[/math], and [math]q[/math] is a polynomial with leading coefficient [math]b[/math]. We can therefore write [math]\lim_{x\to A}r\left(x\right)=B[/math] to describe [math]r[/math]'s end behavior, where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?