Regarding Horizontal and Slant Asymptotes

When the degree is greater in the denominator, then the polynomial fraction is like a proper fraction (such as ) which cannot be converted to a mixed number other than trivially (as "").

For instance, given the following rational function:

...you can't do any long division, because the denominator is of higher degree than is the numerator. The best you can do is to restate the function as:

So, ignoring the fractional portion, you know that the horizontal asymptote is y = 0 (the x-axis), as you can see in the graph below:

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If the degrees of the numerator and the denominator are the same, then the only division you can do is of the leading terms.

For instance, given the following function:

...you can only do one trivial step in the division:

...which means that the original function converts to mixed-number form fairly trivially as:

So, ignoring the fractional part, you know that the horizontal asymptote is y = 2, as you can see in the graph below:

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If the degree is higher on top, then the division gives a polynomial whose degree is the difference between the degrees of the numerator and denominator. Since you'll almost certainly only be doing rationals where the numerator's degree is at most 1 greater than the denominator's degree, then the division will only give you, at most, a linear (straight-line) expression.

For instance, given the following rational function:

...you do the long division:

...and get:

So, ignoring the fractional part, you know that the slant asymptote is y = 2x − 2, as you can see in the graph below:

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In a sense, then, you're always using long division to find the horizontal or slant asymptote. It's just that the long division is explicitly necessary only for finding the slant asymptote.

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