# Mathematical analysis

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Last updated: March 13, 2019

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. x2 – 9 = 0 4×2 = 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: noneA 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. 2 + 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”.

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Since the degree is greater in the denominator than in the numerator, the y-values will be dragged down to the x-axis, and the horizontal asymptote is therefore “y = 0”. Since I have found a horizontal asymptote, I don’t have to look for a slant asymptote. Then the full answer is: domain: all x vertical asymptotes: none horizontal asymptote: y = 0 (the x-axis) slant asymptote: none ________________________________________The Special Case with a “Hole” •Find the domain and all asymptotes of the following function: It so happens that this function can be simplified as: The temptation is to say that y equals x + 1 and therefore that this has no vertical asymptote. But the original function does have a zero in the denominator at x = 2. While the graph of y will look very much like x + 1, it will not quite be the same. Since the degree of the numerator is one greater than the degree of the denominator, I’ll have a slant asymptote (not a horizontal one), and I’ll find that slant asymptote by long division.

Hmm… There wasn’t any remainder when I divided. Actually, that makes sense: since x – 2 is a factor of the numerator and I’m dividing by x – 2, the division should come out evenly. 