Answer:
Step-by-step explanation:
Asymptotes are lines the function approaches, but does not reach, as the independent variable nears some value. They may be horizontal, vertical, or slanted.
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If a rational function has a denominator factor of (x -a) that is not matched by the same factor in the numerator, there will be a vertical asymptote at x=a.
The given function has a denominator factor of x-9 that does not appear in the numerator, so it has a vertical asymptote at x=0.
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As the magnitude of x gets large, the value of a rational function approaches the ratio of the highest-degree terms in the numerator and denominator. Here, that ratio is (3x)/(x) = 3. That is, the given function has a horizontal asymptote at y = 3.
The "end behavior" of the function is that it approaches this asymptote as the value of x approaches ±∞.
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Additional comment
If the degree of the numerator is greater than the degree of the denominator, the quotient of the division of the numerator by the denominator will be an expression of the "slant asymptote" that the function approaches when |x| gets large. The "remainder" from the division will approach zero for large |x|.
We are generally concerned with functions that have a linear "slant asymptote," but the quotient function may have other polynomial behavior, depending on the difference in degrees of the numerator and denominator.
