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Asymptotes

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Asymptotes: Examples (page 4 of 4) 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 = ±…show more content…
By the way, different books may treat this case differently: Some books may say that there is no vertical asymptote where there is a "hole". Check for an example in your text, or ask your instructor. ________________________________________ In general, the procedure for asymptotes is the following: • set the denominator equal to zero and solve o the zeroes (if any) are the vertical asymptotes o everything else is the domain • compare the degrees of the numerator and the denominator o if the degrees are the same, then you have a horizontal asymptote at y = (numerator's leading coefficient) / (denominator's leading coefficient) o if the denominator's degree is greater (by any margin), then you have a horizontal asymptote at y = 0 (the x-axis) o if the numerator's degree is greater (by a margin of 1), then you have a slant asymptote which you will find by doing long division When you're working these problems, try to go through these steps in order, so you can remember them on the test. They're not so hard once you get the hang of them, so be sure to do plenty of practice

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