# Asymptotes

**Topics:**Asymptote, Limit of a function, Rational function

**Pages:**3 (819 words)

**Published:**August 25, 2013

Sections: Vertical asymptotes, Horizontal asymptotes, Slant asymptotes, Examples ________________________________________

In general, you will be given a rational (fractional) function, and you will need to find the domain and any asymptotes. You will need to know what steps to take and how to recognize the different types of asymptotes. •Find the domain and all asymptotes of the following function:

The vertical asymptotes (and any restrictions on the domain) come from the zeroes of the denominator, so I'll set the denominator equal to zero and solve. 4x2 – 9 = 0

4x2 = 9

x2 = 9/4

x = ± 3/2

Then the domain is all x-values other than ± 3/2, and the two vertical asymptotes are at x = ± 3/2.

Since the degrees of the numerator and the denominator are the same (each being 2), then this rational has a non-zero (non-x-axis) horizontal asymptote, and does not have a slant asymptote; the horizontal asymptote is found by dividing the leading terms:

Then the full answer is:

domain:

vertical asymptotes: x = ± 3/2

horizontal asymptote: y = 1/4

slant asymptote: none

A given rational function may or may not have a vertical asymptote (depending upon whether the denominator ever equals zero), but it will always have either a horizontal or else a slant asymptote. Note, however, that the function will only have one of these two; you will have either a horizontal asymptote or else a slant asymptote, but not both. As soon as you see that you'll have one of them, don't bother looking for the other one. •Find the domain and all asymptotes of the following function:

The vertical asymptotes come from the zeroes of the denominator, so I'll set the denominator equal to zero and solve. x2 + 9 = 0

x2 = –9

Oops! This has no solution. Since the denominator has no zeroes, then there are no vertical asymptotes and the domain is "all x". Since the degree is greater in the denominator than in the numerator, the y-values...

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