# A close look at end behavior

- Author:
- Charlie Barnes

My problem with BOB0 - BOTN - EATS DC is two-fold. Like most mnemonic devices in K-12 mathematics:

- It oversimplifies the concept.
- It is a technique that encourages rote memorization, rather than conceptual understanding.

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I used polynomial long division there. So the in is called the*quotient*, and the is called the*remainder*(because goes into times with a remainder of ). Since , , and thus our rational function has a horizontal asymptote at .Which type of rational function was this?

What is the quotient in a BOB0 rational function? Explain why that's consistent with the BOB0 rule.

Here's an example that begins to illustrate why I think BOB0 - BOTN - EATS DC oversimplifies this concept:

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So the quotient is . Use the graph below to plot this rational function.What you're seeing is that this function is asymptotic not to a horizontal line, but to the line . is what's called a *slant asymptote* of . Explain how BOTN oversimplifies this situation.

## Plot a rational function that is asymptotic to a parabola.

Plot a rational function that is (not equal but) asymptotic to the parabola . (Hint: You'll want to work backwards through the polynomial long division. Finding a common denominator will be a helpful technique.)