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MHF4U: Advanced Functions
Oblique Asymptotes
J. Garvin
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Oblique Asymptotes
In addition to horizontal and vertical asymptotes, a function may have oblique asymptotes. Oblique asymptotes are sometimes called โslantโ asymptotes because they have the form y = ax + b, where a 6 = 0. A function will have an oblique asymptote if the degree of the numerator is one greater than that of the denominator. A function will never have both oblique and horizontal asymptotes.
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Oblique Asymptotes
Example Determine the equation of the oblique asymptote for f (x) = x
x , and graph the function.
Use long division to determine the equation of the oblique asymptote.
x x)^ x^2 โ 1 โ x^2
The equation of the oblique asymptote is y = x. J. Garvin โ Oblique AsymptotesSlide 3/
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Oblique Asymptotes
There is a vertical asymptote at x = 0, as given by the denominator. There is no f (x)-intercept, since setting x = 0 causes a division by zero error. Since x^2 โ 1 is a difference of squares, f (x) has x-intercepts at ยฑ1. These points, along with the asymptotes, give us enough information to accurately sketch the graph, but we can test values of x close to the vertical asymptote to get a better picture. As x โ 0 from the left, f (x) โ โ, and as x โ 0 from the right, f (x) โ โโ.
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Oblique Asymptotes
Example Graph f (x) = x
(^2) + 2x โ 3 x + 1.
x + 1 x + 1
x^2 + 2x โ 3 โ x^2 โ x x โ 3 โ x โ 1 โ 4
There is an oblique asymptote with equation y = x + 1.
Oblique Asymptotes
There is a vertical asymptote at x = โ1, since the denominator is zero when x = โ1. Setting x = 0 gives an f (x)-intercept of โ3.
Since f (x) factors as f (x) = (x^ โ x^ 1)( + 1x^ + 3) , there are x-intercepts at 1 and โ3. As x โ 0 โ, f (x) โ โ, and as x โ 0 +, f (x) โ โโ. In the notation above, x โ kโ^ means โas x approaches k from the leftโ, while x โ k+^ is from the right.
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Oblique Asymptotes
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Example
Graph f (x) = x
(^2) + 2x x + 2.
In this case, note that f (x) = x( xx + 2+ 2) = x, where x 6 = โ2.
Thus, the graph of f (x) is the same as the graph of y = x, but with a point discontinuity at x = โ2. It is generally a good idea to check for the same root in both the numerator and denominator before doing any extra work.
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A More Complex Example
Graph f (x) = x
(^3) โ 2 x (^2) โ x + 2 x^2 โ x โ 6.
Let f (x) = g h^ ((xx)).
By the FT, g (1) = g (2) = g (โ1) = 0, and h(โ2) = h(3) = 0.
Therefore, f (x) = (x^ โ(x^ 1)( + 2)(x^ โ^ x2)( โx 3)^ + 1).
There are vertical asymptotes at x = โ2 and x = 3. The x-intercepts are at 1, 2 and โ1. The f (x)-intercept is at โ 13.
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Oblique Asymptotes
Use long division to determine the equation of the oblique asymptote. x โ 1 x^2 โ x โ 6 )^ x^3 โ 2 x^2 โ x + 2 โ x^3 + x^2 + 6x โ x^2 + 5x + 2 x^2 โ x โ 6 4 x โ 4 There is an oblique asymptote with equation y = x โ 1.
